"use strict"; const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = { a: "a", annotation: "annotation", blockquote: "blockquote", code: "code", details: "details", div: "div", em: "em", h2: "h2", h3: "h3", h4: "h4", h5: "h5", img: "img", li: "li", math: "math", mi: "mi", mn: "mn", mo: "mo", mrow: "mrow", mstyle: "mstyle", msub: "msub", msubsup: "msubsup", msup: "msup", mtable: "mtable", mtd: "mtd", mtext: "mtext", mtr: "mtr", ol: "ol", p: "p", path: "path", pre: "pre", semantics: "semantics", span: "span", strong: "strong", summary: "summary", svg: "svg", table: "table", tbody: "tbody", td: "td", th: "th", thead: "thead", tr: "tr", ul: "ul", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsxs(_components.div, { "data-callout": true, "data-callout-type": "tip", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "【复习提示】" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, {}), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "本章内容多以选择题或综合题的形式考察，但统考也会出涉及树遍历相关的算法题。" }), "\n", _jsx(_components.li, { children: "数和二叉树的性质、遍历操作、转换、存储结构和操作特性等。" }), "\n", _jsx(_components.li, { children: "满二叉树、完全二叉树、线索二叉树、哈夫曼树等定义和性质，都是选择题必然会涉及的内容。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "51-树的基本概念", children: [_jsx(_components.a, { className: "heading-link", href: "#51-树的基本概念", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5.1 树的基本概念"] }), "\n", _jsxs(_components.h3, { id: "511-树的定义", children: [_jsx(_components.a, { className: "heading-link", href: "#511-树的定义", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5.1.1 树的定义"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "定义" }), "：树是 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "≥" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n(n\\geq 0)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≥" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " 个结点的有限集。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["当 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n=0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }), " 时，称为", _jsx(_components.strong, { children: "空树" }), "。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["在任意一棵非空树中应满足：", "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["有且仅有一个特定的称为", _jsx(_components.strong, { children: "根" }), "的结点。"] }), "\n", _jsxs(_components.li, { children: ["当 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n>1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5782em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), " 时，其余结点可分为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m(m>0)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " 个互不相交的有限集 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mo, { children: "…" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "m" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_{1},T_{2},\\dots,T_{m}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "minner", children: "…" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), "，其中每个集合本身又是一颗树，并且称为根的", _jsx(_components.strong, { children: "子树" }), "。"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["显然树的定义是递归的，即在树的定义中又用到了其自身，树是一种", _jsx(_components.strong, { children: "递归的数据结构" }), "。"] }), "\n", _jsx(_components.p, { children: "树作为一种逻辑结构，同时也是一种分层结构，具有以下两个特点：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.p, { children: "树的根结点没有前驱，除根结点外的所有结点有且只有一个前驱。" }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.p, { children: "树中所有结点都可以有零个或多个后继。" }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: ["树适用于表示具有", _jsx(_components.em, { children: "层次结构" }), "的数据。"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: ["树中的某个结点（除根结点外）最多只和上一层的一个结点（其父结点）有直接关系，根结点没有直接上层结点，因此在 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个结点的树中有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n-1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), " 条边。"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.p, { children: "而树中每个结点与其下一层的零个或多个结点（其孩子结点）都有直接关系。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "512-基本术语", children: [_jsx(_components.a, { className: "heading-link", href: "#512-基本术语", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5.1.2 基本术语"] }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202511291355091.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["祖先、子孙、双亲、孩子、兄弟和堂兄弟。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "祖先" }), "：考虑结点K，从根A到结点K的唯一路径上的所有其他结点，称为K的", _jsx(_components.strong, { children: "祖先&子孙" }), "。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["如结点B是结点K的", _jsx(_components.em, { children: "祖先" }), "，而K是B的", _jsx(_components.em, { children: "子孙" }), "，结点B的子孙包括E，F，K，L。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "双亲&孩子" }), "：路径上最接近结点K的结点E称为K的", _jsx(_components.em, { children: "双亲" }), "，而K为E的", _jsx(_components.em, { children: "孩子" }), "。", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "根A是树中唯一没有双亲的结点。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "兄弟" }), "：有相同双亲的结点称为", _jsx(_components.em, { children: "兄弟" }), "，如结点K和结点L有相同的双亲E，即K和L为兄弟。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "堂兄弟" }), "：双亲在同一层的结点互为", _jsx(_components.em, { children: "堂兄弟" }), "，结点G与E，F，H，I，J互为堂兄弟。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["结点的层次、深度和高度。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "结点的层次" }), "：结点的层次从树根开始定义，根结点为第1层，它的孩子为第2层，以此类推。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "结点的深度" }), "：就是结点所在的层次。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "树的高度（或深度）" }), " ：是树中结点的最大层数。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "结点的高度" }), "：是以该结点为根的子树的高度。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["结点的度和树的度。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "结点的度" }), "：树中一个结点的", _jsx(_components.em, { children: "孩子的个数" }), "称为该", _jsx(_components.em, { children: "结点的度" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "数的度" }), "：树中结点的最大度数称为", _jsx(_components.em, { children: "树的度" }), "。如结点B的度为2，结点D的度为3，树的度为3。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["分支结点和叶结点。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "分支结点" }), "：度大于0的结点称为", _jsx(_components.em, { children: "分支结点" }), "（也称非终端结点）；"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "叶结点" }), "：度为0（没有孩子结点）的结点称为", _jsx(_components.em, { children: "叶结点" }), "（也称终端结点）。", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "在分支结点中，每个结点的分支数就是该结点的度。" }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["有序树和无序树", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "有序树&无序树" }), "：树中结点的各子树从左到右是有次序的，不能交换，称该树为", _jsx(_components.em, { children: "有序树" }), "，否则称为", _jsx(_components.em, { children: "无序树" }), "。\n假设图5.1是有序树，若将子结点位置互换，则变成一颗不同的树。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["路径和路径长度。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "路径" }), "：树中两个结点之间的", _jsx(_components.em, { children: "路径" }), "是由这两个结点之间所经过的结点序列构成的。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "路径长度" }), "：", _jsx(_components.em, { children: "路径长度" }), "是路径上所经过边的个数。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "树的路径长度" }), "：所有结点路径长度总和。（课后题有，但是王道书知识点里却不讲，令人语塞）"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["森林。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "森林" }), "：", _jsx(_components.em, { children: "森林" }), "是 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "≥" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m(m\\geq0)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≥" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " 棵互不相交的", _jsx(_components.em, { children: "树的集合" }), "。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "森林" }), "的概念与树的概念十分接近，因为只要把树的根结点删去就成了森林。"] }), "\n", _jsxs(_components.li, { children: ["反之，只要给 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "m" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" })] }) })] }), " 棵独立的树加上一个结点，并把这 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "m" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" })] }) })] }), " 棵树作为该结点的子树，则森林就变成了树。"] }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "上述概念无需刻意记忆，根据实际例子理解即可。考研时不大可能直接考概念，而都是结构具体的题目考查，做题时，遇到不熟悉的大可翻书，练习的多了自然就记住了。" }), "\n"] }), "\n", _jsxs(_components.h3, { id: "513-树的性质", children: [_jsx(_components.a, { className: "heading-link", href: "#513-树的性质", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5.1.3 树的性质"] }), "\n", _jsx(_components.p, { children: "树的基本性质如下：" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["数的结点数 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 等于所有结点的度数之和加 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "1" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "1" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] }) })] }), ".", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["结点的度是指该结点的孩子数量，每个结点与其每个孩子都由唯一的边相连，因此树中", _jsx(_components.strong, { children: "所有结点的度数之和等于树中的边数之和。" })] }), "\n", _jsxs(_components.li, { children: ["树中的结点（除根外）都有唯一的双亲，因此结点树 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 等于边树之和加1，即所有结点的度数之和加1."] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "m" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" })] }) })] }), " 的树中第 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 层上至多有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "m" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m^{i-1}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8247em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8247em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] })] }) })] }), " 个结点 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "≥" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(i\\geq 1)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≥" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), "。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["第1层至多有1个结点（根结点），第2层至多有m个结点，第3层至多有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mn, { children: "2" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m^{2}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }) }) }) })] })] }) })] }), " 个结点，以此类推。使用数学归纳法可推出第 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 层至多有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "m" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m^{i-1}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8247em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8247em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] })] }) })] }), " 个结点。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["高度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "h" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "h" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "h" })] }) })] }), " 的 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "m" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" })] }) })] }), " 叉树至多有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "h" })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(m^{h}-1)/(m-1)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0991em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8491em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "h" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mord", children: "/" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " 个结点。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["当各层结点数达到最大时，树中至多有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mo, { children: "⋯" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "m" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "h" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "h" })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "1+m+m^{2}+\\dots+m^{h-1}=(m^{h}-1) / (m-1)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8974em", verticalAlign: "-0.0833em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "minner", children: "⋯" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8491em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8491em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "h" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0991em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8491em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "h" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mord", children: "/" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " 个结点。"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "等比数列的求和公式。" }), "\n"] }), "\n", _jsxs(_components.ol, { start: "4", children: ["\n", _jsxs(_components.li, { children: ["度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "m" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" })] }) })] }), "、具有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个结点的树的最"] }), "\n", _jsxs(_components.li, { children: ["小高度 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "h" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "h" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "h" })] }) })] }), " 为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "⌈" }), _jsxs(_components.msub, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "log" }), _jsx(_components.mo, { children: "⁡" })] }), _jsx(_components.mi, { children: "m" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { stretchy: "false", children: "⌉" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lceil \\log_{m}(n(m-1)+1) \\rceil" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "⌈" }), _jsxs(_components.span, { className: "mop", children: [_jsxs(_components.span, { className: "mop", children: ["lo", _jsx(_components.span, { style: { marginRight: "0.0139em" }, children: "g" })] }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.0573em" }, children: _jsxs(_components.span, { style: { top: "-2.4559em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2441em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")⌉" })] })] })] }), "。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["为了使树的高度最小，在前 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "h" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "h-1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7778em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "h" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), " 层中，每层的结点数都要达到最大，前 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "h" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "h-1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7778em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "h" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), " 层最多有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "m" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "h" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(m^{h-1}-1) / (m-1)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0991em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8491em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "h" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mord", children: "/" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " 个结点，前 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "h" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "h" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "h" })] }) })] }), " 层最多有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "h" })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(m^{h}-1)/(m-1)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0991em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8491em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "h" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mord", children: "/" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " 个结点。"] }), "\n", _jsxs(_components.li, { children: ["因此 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "m" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "h" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "≤" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "h" })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(m^{h-1}-1) / (m-1)" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n>1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5782em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), " 时，结点 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 当双亲结点的编号为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "⌊" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { stretchy: "false", children: "⌋" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lfloor i / 2 \\rfloor" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "⌊" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mord", children: "/2" }), _jsx(_components.span, { className: "mclose", children: "⌋" })] }) })] }), "。"] }), "\n", _jsxs(_components.li, { children: ["若结点 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 有左、右孩子，则左孩子编号为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), "，右孩子编号为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2i+1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7429em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "。"] }), "\n", _jsxs(_components.li, { children: ["结点 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 所在层次（深度）为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "⌊" }), _jsxs(_components.msub, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "log" }), _jsx(_components.mo, { children: "⁡" })] }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "⌋" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lfloor \\log_2i \\rfloor+1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "⌊" }), _jsxs(_components.span, { className: "mop", children: [_jsxs(_components.span, { className: "mop", children: ["lo", _jsx(_components.span, { style: { marginRight: "0.0139em" }, children: "g" })] }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.207em" }, children: _jsxs(_components.span, { style: { top: "-2.4559em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2441em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "⌋" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["具有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(n> 0)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " 结点的", _jsx(_components.strong, { children: "完全二叉树" }), "的高度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "⌈" }), _jsxs(_components.msub, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "log" }), _jsx(_components.mo, { children: "⁡" })] }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { stretchy: "false", children: "⌉" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lceil \\log_{2}(n+1) \\rceil" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "⌈" }), _jsxs(_components.span, { className: "mop", children: [_jsxs(_components.span, { className: "mop", children: ["lo", _jsx(_components.span, { style: { marginRight: "0.0139em" }, children: "g" })] }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.207em" }, children: _jsxs(_components.span, { style: { top: "-2.4559em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2441em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")⌉" })] })] })] }), " 或 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "⌊" }), _jsxs(_components.msub, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "log" }), _jsx(_components.mo, { children: "⁡" })] }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: "⌋" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lfloor \\log_{2}n \\rfloor+1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "⌊" }), _jsxs(_components.span, { className: "mop", children: [_jsxs(_components.span, { className: "mop", children: ["lo", _jsx(_components.span, { style: { marginRight: "0.0139em" }, children: "g" })] }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.207em" }, children: _jsxs(_components.span, { style: { top: "-2.4559em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2441em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: "⌋" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["设高度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "h" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "h" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "h" })] }) })] }), "，根据性质3和完全二叉树的定义有：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "h" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "≤" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "h" })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2^{h-1}-1" }), "lchild", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 递归遍历左子树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "PreOrder" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 递归遍历右子树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "瞧瞧这优雅的递归代码。" }), "\n"] }), "\n", _jsxs(_components.h4, { id: "2-中序遍历inorder", children: [_jsx(_components.a, { className: "heading-link", href: "#2-中序遍历inorder", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "2. 中序遍历（InOrder）"] }), "\n", _jsx(_components.p, { children: "若二叉树为空，则什么也不做；否则，" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "中序遍历左子树" }), "\n", _jsx(_components.li, { children: "访问根结点" }), "\n", _jsx(_components.li, { children: "中序遍历右子树" }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "tip", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "命题追踪" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "中序序列中结点关系的分析" }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "图5.8中的虚线表示对该二叉树进行中序遍历的路径，得到中序遍历序列为264135。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512022153436.png", alt: "image.png", className: "internal" }), "\n", _jsx(_components.p, { children: "对应的递归算法如下：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "InOrder" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "BiTree T", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token operator", children: "!=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "InOrder" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 递归遍历左子树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "visit" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 访问根结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "InOrder" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 递归遍历右子树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.h4, { id: "3-后序遍历postorder", children: [_jsx(_components.a, { className: "heading-link", href: "#3-后序遍历postorder", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "3. 后序遍历（PostOrder）"] }), "\n", _jsx(_components.p, { children: "若二叉树为空，则什么都不做，否则，" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "后序遍历左子树" }), "\n", _jsx(_components.li, { children: "后序遍历右子树" }), "\n", _jsx(_components.li, { children: "访问根结点" }), "\n"] }), "\n", _jsx(_components.p, { children: "图5.9 中的虚线表示对二叉树进行后序遍历的路径，得到后序遍历序列为642531。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512022201039.png", alt: "image.png", className: "internal" }), "\n", _jsx(_components.p, { children: "对应的递归算法如下：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "PostOrder" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "BiTree T", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token operator", children: "!=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "PostOrder" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 递归遍历左子树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "PostOrder" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 递归遍历右子树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "visit" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 访问根结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["我们对比这三个代码不难发现，唯一的区别就是 ", _jsx(_components.code, { className: "language-auto", children: "visit(T);" }), " 的相对位置而已。"] }), "\n"] }), "\n", _jsx(_components.p, { children: "上述三种遍历算法中，递归遍历左、右子树的顺序都是固定的，只是访问根结点的顺序不同。" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["不管采用哪种方法，", _jsx(_components.strong, { children: "每个结点都访问一次且仅访问一次" }), "，所以时间复杂度都是 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。"] }), "\n", _jsxs(_components.li, { children: ["在递归遍历中，递归工作栈的栈深恰好为树的深度，所以在最坏情况下，二叉树是有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个结点且深度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 的单支树，遍历算法的空间复杂度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "最坏也就是把一整颗树都压入栈中而已。" }), "\n"] }), "\n", _jsxs(_components.h4, { id: "4-层次遍历", children: [_jsx(_components.a, { className: "heading-link", href: "#4-层次遍历", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "4. 层次遍历"] }), "\n", _jsx(_components.p, { children: "图5.10所示为二叉树的层次遍历，即按照箭头所指的方向，按照1，2，3，4的层次顺序，自上而下、从左至右对二叉树中的各个结点进行逐层访问。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512022207245.png", alt: "image.png", className: "internal" }), "\n", _jsx(_components.p, { children: "进行层次遍历时，需要借助一个队列。层次遍历的思想如下：" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["首先将", _jsx(_components.strong, { children: "根结点入队" }), "。"] }), "\n", _jsxs(_components.li, { children: ["若", _jsx(_components.strong, { children: "队列非空" }), "，则", _jsx(_components.em, { children: "队头结点出队" }), "，", _jsx(_components.em, { children: "访问该结点" }), "，", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["若它", _jsx(_components.strong, { children: "有左孩子" }), "，则将其左孩子", _jsx(_components.em, { children: "入队" }), "；"] }), "\n", _jsxs(_components.li, { children: ["若它", _jsx(_components.strong, { children: "有右孩子" }), "，则将其右孩子", _jsx(_components.em, { children: "入队" }), "。"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.li, { children: "重复步骤（2.），直至队列为空。" }), "\n"] }), "\n", _jsx(_components.p, { children: "二叉树的层次遍历算法如下：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "LevelOrder" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "BiTree T", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "InitQueue" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 初始化辅助队列" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" BiTree p", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "EnQueue" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "," }), "T", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 将根结点入队" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "while" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token operator", children: "!" }), _jsx(_components.span, { className: "token function", children: "IsEmpty" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), " ", _jsx(_components.span, { className: "token comment", children: "// 队列不为空则继续循环" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "DeQueue" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "," }), " P", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 队头结点出队" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "visit" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 访问出队结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild", _jsx(_components.span, { className: "token operator", children: "!=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "EnQueue" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "," }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 若左孩子不为空，则左孩子入队" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token operator", children: "!=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "EnQueue" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Q", _jsx(_components.span, { className: "token punctuation", children: "," }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 若右孩子不为空，则右孩子入队" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.details, { "data-callout": true, "data-callout-type": "abstract-", children: ["\n", _jsx(_components.summary, { "data-callout-title": true, children: "王道这里的算法不太方便自己实现，更推荐hello算法上的。" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, {}), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "const" }), " ", _jsx(_components.span, { className: "token keyword", children: "int" }), " MAX_SIZE ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "100" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: "*" }), " ", _jsx(_components.span, { className: "token function", children: "LevelOrder" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Node ", _jsx(_components.span, { className: "token operator", children: "*" }), "root", _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token keyword", children: "int" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), "size", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "int" }), " front", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " rear", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "int" }), " index", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), "arr", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Node ", _jsx(_components.span, { className: "token operator", children: "*" }), "node", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" Node ", _jsx(_components.span, { className: "token operator", children: "*" }), _jsx(_components.span, { className: "token operator", children: "*" }), "queue", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 这里特别说明一下，这是一个指向queue的数组，数组中的每个元素都是*queue指针。" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 辅助队列" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" queue ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "Node ", _jsx(_components.span, { className: "token operator", children: "*" }), _jsx(_components.span, { className: "token operator", children: "*" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token function", children: "malloc" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "sizeof" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Node ", _jsx(_components.span, { className: "token operator", children: "*" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), " MAX_SIZE", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 层序遍历保存" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" arr ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: "*" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token function", children: "malloc" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "sizeof" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), " MAX_SIZE", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [" queue", _jsx(_components.span, { className: "token punctuation", children: "[" }), "rear", _jsx(_components.span, { className: "token operator", children: "++" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " root", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 现将头结点入队" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "while" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "front ", _jsx(_components.span, { className: "token operator", children: "<" }), " rear", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 当front" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 队头出队" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" node ", _jsx(_components.span, { className: "token operator", children: "=" }), " queue", _jsx(_components.span, { className: "token punctuation", children: "[" }), "front", _jsx(_components.span, { className: "token operator", children: "++" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 复制到数组" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" arr", _jsx(_components.span, { className: "token punctuation", children: "[" }), "index", _jsx(_components.span, { className: "token operator", children: "++" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " node", _jsx(_components.span, { className: "token operator", children: "->" }), "data", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "node", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild ", _jsx(_components.span, { className: "token operator", children: "!=" }), " ", _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" queue", _jsx(_components.span, { className: "token punctuation", children: "[" }), "rear", _jsx(_components.span, { className: "token operator", children: "++" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " node", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "node", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild ", _jsx(_components.span, { className: "token operator", children: "!=" }), " ", _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" queue", _jsx(_components.span, { className: "token punctuation", children: "[" }), "rear", _jsx(_components.span, { className: "token operator", children: "++" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " node", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token operator", children: "*" }), "size ", _jsx(_components.span, { className: "token operator", children: "=" }), " index", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" arr ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token operator", children: "*" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token function", children: "realloc" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "arr", _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token keyword", children: "sizeof" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token operator", children: "*" }), "size", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "free" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "queue", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 释放辅助队列" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " arr", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "在复习过程中，读者应将上述二叉树层次遍历的算法作为一个模版，熟练掌握其执行过程，并达到熟练手写的程度。" }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "attention", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "注意" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "遍历是二叉树各种操作的基础，例如对于一棵给定二叉树：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "求结点的双亲" }), "\n", _jsx(_components.li, { children: "求结点的孩子" }), "\n", _jsx(_components.li, { children: "求二叉树的深度" }), "\n", _jsx(_components.li, { children: "求叶结点个数" }), "\n", _jsx(_components.li, { children: "判断两棵二叉树是否相同" }), "\n", _jsx(_components.li, { children: "…\n所有这些操作都是在遍历的过程中进行的，因此必须掌握二叉树的各种遍历过程，并能灵活运用以解决各种问题。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h4, { id: "5-由遍历序列构造二叉树", children: [_jsx(_components.a, { className: "heading-link", href: "#5-由遍历序列构造二叉树", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5. 由遍历序列构造二叉树"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "tip", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "命题追踪" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "先序序列对应的不同二叉树的分析" }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "对于一棵给定的二叉树，其：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "先序序列" }), "\n", _jsx(_components.li, { children: "中序序列" }), "\n", _jsx(_components.li, { children: "后序序列" }), "\n", _jsx(_components.li, { children: "层序序列\n都是确定的。然而，只给出其中的任意一种不能唯一地确定一棵二叉树。若已知中序序列，再给出其他三种遍历序列中的任意一种，就可以唯一地确定一二叉树。" }), "\n"] }), "\n", _jsxs(_components.h5, { id: "1由先序序列和中序序列构造二叉树", children: [_jsx(_components.a, { className: "heading-link", href: "#1由先序序列和中序序列构造二叉树", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "（1）由先序序列和中序序列构造二叉树"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "tip", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "命题追踪" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "先序序列和中序序列相同时确定的二叉树\n由先序遍历和中序遍历构造一棵二叉树" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["在", _jsx(_components.strong, { children: "先序序列" }), "中，第一个结点一定是二叉树的根结点；"] }), "\n", _jsxs(_components.li, { children: ["而在", _jsx(_components.strong, { children: "中序遍历" }), "中，根结点必然将中序序列分割成两个子序列，", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "前一个子序列是根的左子树的中序序列，" }), "\n", _jsx(_components.li, { children: "后一个子序列是根的右子树的中序序列。" }), "\n"] }), "\n"] }), "\n", _jsx(_components.li, { children: "左子树的中序序列和先序序列的长度是相等的，右子树的中序序列和先序序列的长度是相等的。" }), "\n"] }), "\n", _jsx(_components.p, { children: "根据这两个子序列，可以在先序序列中找到左子树的先序序列和右子树的先序序列，如图5.11所示。" }), "\n", _jsx(_components.p, { children: "如此递归地分解下去，便能唯一地确定这棵二叉树。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512032142546.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.p, { children: ["例如，求先序序列（", _jsx(_components.em, { children: "ABCDEFGHI" }), "）和中序序列（", _jsx(_components.em, { children: "BCAEDGHFI" }), "）所确定的二叉树。"] }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["首先，由先序序列可知：", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "A" }), "为二叉树的根结点。"] }), "\n", _jsxs(_components.li, { children: ["中序序列中", _jsx(_components.em, { children: "A" }), "之前的", _jsx(_components.em, { children: "BC" }), "为左子树的中序序列，"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "EDGHFI" }), "为右子树的中序序列。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["然后，由先序序列可知：", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "B" }), "是左子树的根结点，"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "D" }), "是右子树的根结点。"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "以此类推，就能将剩下的结点继续分解下去，最后得到的二叉树如图5.12（c）所示。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512032147996.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.h5, { id: "2由后序序列和中序序列构造二叉树", children: [_jsx(_components.a, { className: "heading-link", href: "#2由后序序列和中序序列构造二叉树", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "（2）由后序序列和中序序列构造二叉树"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "tip", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "命题追踪" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "由后序遍历和树形构造一棵二叉树" }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "同理，由二叉树的后序遍历和中序遍历也可以唯一地确定一棵二叉树。因为后序序列的最后一个结点就如同先序序列的第一个结点，可以将中序序列分割成两个子序列，如图5.13所示，然后用类似的方法递归地进行分解，进而唯一地确定这棵二叉树。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512032152070.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.p, { children: ["请读者分析后序序列（", _jsx(_components.em, { children: "CBEHGIFDA" }), "）和中序序列（", _jsx(_components.em, { children: "BCAEDGHFI" }), "）所确定的二叉树。"] }), "\n", _jsxs(_components.h5, { id: "3由层序序列和中序序列构造二叉树", children: [_jsx(_components.a, { className: "heading-link", href: "#3由层序序列和中序序列构造二叉树", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "（3）由层序序列和中序序列构造二叉树"] }), "\n", _jsx(_components.p, { children: "在层序遍历中，第一个结点一定是二叉树的根结点，这样就将中序序列分割成了左子树的中序序列和右子树的中序序列。" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "若存在左子树，则层序序列的第二个结点一定是左子树的根，可进一步划分左子树；" }), "\n", _jsx(_components.li, { children: "若存在右子树，则层序序列中紧接着的下一个结点一定是右子树的根，可进一步划分右子树，如图5.14所示。" }), "\n"] }), "\n", _jsx(_components.p, { children: "采用这种方法继续分解下去就能唯一确定这棵二叉树。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512032156170.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.p, { children: ["请自行分析层序序列（", _jsx(_components.em, { children: "ABDCEFGIH" }), "）和中序遍历（", _jsx(_components.em, { children: "BCAEDGHFI" }), "）所确定的二叉树。"] }), "\n", _jsxs(_components.p, { children: ["注意，先序序列、后序序列和层序序列的两两组合，无法唯一确定一棵二叉树。如图5.15所示的两棵二叉树的先序序列都为", _jsx(_components.em, { children: "AB" }), "，后序序列都为", _jsx(_components.em, { children: "BA" }), "，层序序列都为", _jsx(_components.em, { children: "AB" }), "。"] }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512032159349.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.h3, { id: "532-线索二叉树", children: [_jsx(_components.a, { className: "heading-link", href: "#532-线索二叉树", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5.3.2 线索二叉树"] }), "\n", _jsxs(_components.h4, { id: "1-线索二叉树的基本概念", children: [_jsx(_components.a, { className: "heading-link", href: "#1-线索二叉树的基本概念", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "1. 线索二叉树的基本概念"] }), "\n", _jsx(_components.p, { children: "遍历二叉树是以一定的规则将二叉树中的结点排列成一个线性序列，从而得到几种遍历序列，使得该序列中的每个结点（第一个和最后一个除外）都有一个直接前驱和直接后继。" }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "tip", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "命题追踪" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "线索二叉树的定义" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["传统的二叉链表存储仅能体现一种父子关系，不能直接得到结点在遍历中的前驱或后继。前面提到，在含", _jsx(_components.em, { children: "n" }), "个结点的二叉树中，有", _jsx(_components.em, { children: "n+1" }), "个空指针。这是因为每个叶结点都有2个空指针，每个度为1的结点都有1个空指针，空指针总数为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "2" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2n_{0}+n_{1}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7944em", verticalAlign: "-0.15em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] }), "，又 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n_{0}=n_{2}+1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "，所以空指针总数为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n_{0}+n_{1}+n_{2}+1=n+ 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "。"] }), "\n", _jsx(_components.p, { children: "由此设想能否利用这些空指针来存放指向其前驱或后继的指针？这样就可以像遍历单链表那样方便地遍历二叉树。" }), "\n", _jsx(_components.p, { children: "引入线索二叉树正是为了加快查找结点前驱和后继的速度。" }), "\n", _jsx(_components.p, { children: "规定：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["若无左子树，令", _jsx(_components.em, { children: "lchild" }), "指向其前驱结点；"] }), "\n", _jsxs(_components.li, { children: ["若无右子树，令", _jsx(_components.em, { children: "rchild" }), "指向其后继结点。"] }), "\n"] }), "\n", _jsx(_components.p, { children: "如图5.16所示，还需要增加两个标志域，以标识指针域指向左（右）孩子或前驱（后继）。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512032215123.png", alt: "image.png", className: "internal" }), "\n", _jsx(_components.p, { children: "其中标志域的含义如下：" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mtable, { rowspacing: "0.25em", columnalign: "right", columnspacing: "", children: [_jsxs(_components.mtr, { children: [_jsx(_components.mtd, { className: "mtr-glue" }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "true", children: _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "l" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "g" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "{" }), _jsxs(_components.mtable, { rowspacing: "0.36em", columnalign: "left left", columnspacing: "1em", children: [_jsxs(_components.mtr, { children: [_jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { separator: "true", children: "," })] }) }) }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mtext, { children: "lchild域指示结点的左孩子" }), _jsx(_components.mo, { separator: "true", children: "," })] }) }) })] }), _jsxs(_components.mtr, { children: [_jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { separator: "true", children: "," })] }) }) }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsx(_components.mtext, { children: "lchild域指示结点的前驱" }) }) })] })] })] })] }) }) }), _jsx(_components.mtd, { className: "mtr-glue" }), _jsx(_components.mtd, { className: "mml-eqn-num" })] }), _jsxs(_components.mtr, { children: [_jsx(_components.mtd, { className: "mtr-glue" }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "true", children: _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "g" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "{" }), _jsxs(_components.mtable, { rowspacing: "0.36em", columnalign: "left left", columnspacing: "1em", children: [_jsxs(_components.mtr, { children: [_jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { separator: "true", children: "," })] }) }) }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mtext, { children: "rchild域指示结点的右孩子" }), _jsx(_components.mo, { separator: "true", children: "," })] }) }) })] }), _jsxs(_components.mtr, { children: [_jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { separator: "true", children: "," })] }) }) }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsx(_components.mtext, { children: "rchild域指示结点的后继" }) }) })] })] })] })] }) }) }), _jsx(_components.mtd, { className: "mtr-glue" }), _jsx(_components.mtd, { className: "mml-eqn-num" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\begin{align}\nltag=\\begin{cases}\n0, & \\text{lchild域指示结点的左孩子}, \\\\\n1, & \\text{lchild域指示结点的前驱}\n\\end{cases} \\\\\nrtag=\\begin{cases}\n0, & \\text{rchild域指示结点的右孩子}, \\\\\n1, & \\text{rchild域指示结点的后继}\n\\end{cases}\n\\end{align}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "6.3001em", verticalAlign: "-2.9em" } }), _jsx(_components.span, { className: "mtable", children: _jsx(_components.span, { className: "col-align-r", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "3.4em" }, children: [_jsxs(_components.span, { style: { top: "-5.4em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.75em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0197em" }, children: "l" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "g" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size4", children: "{" }) }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mtable", children: [_jsx(_components.span, { className: "col-align-l", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.69em" }, children: [_jsxs(_components.span, { style: { top: "-3.69em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mpunct", children: "," })] })] }), _jsxs(_components.span, { style: { top: "-2.25em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mpunct", children: "," })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.19em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "arraycolsep", style: { width: "1em" } }), _jsx(_components.span, { className: "col-align-l", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.69em" }, children: [_jsxs(_components.span, { style: { top: "-3.69em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsxs(_components.span, { className: "mord text", children: [_jsx(_components.span, { className: "mord", children: "lchild" }), _jsx(_components.span, { className: "mord cjk_fallback", children: "域指示结点的左孩子" })] }), _jsx(_components.span, { className: "mpunct", children: "," })] })] }), _jsxs(_components.span, { style: { top: "-2.25em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord text", children: [_jsx(_components.span, { className: "mord", children: "lchild" }), _jsx(_components.span, { className: "mord cjk_fallback", children: "域指示结点的前驱" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.19em" }, children: _jsx(_components.span, {}) }) })] }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] }), _jsxs(_components.span, { style: { top: "-2.1em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.75em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "r" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "g" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size4", children: "{" }) }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mtable", children: [_jsx(_components.span, { className: "col-align-l", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.69em" }, children: [_jsxs(_components.span, { style: { top: "-3.69em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mpunct", children: "," })] })] }), _jsxs(_components.span, { style: { top: "-2.25em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mpunct", children: "," })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.19em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "arraycolsep", style: { width: "1em" } }), _jsx(_components.span, { className: "col-align-l", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.69em" }, children: [_jsxs(_components.span, { style: { top: "-3.69em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsxs(_components.span, { className: "mord text", children: [_jsx(_components.span, { className: "mord", children: "rchild" }), _jsx(_components.span, { className: "mord cjk_fallback", children: "域指示结点的右孩子" })] }), _jsx(_components.span, { className: "mpunct", children: "," })] })] }), _jsxs(_components.span, { style: { top: "-2.25em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord text", children: [_jsx(_components.span, { className: "mord", children: "rchild" }), _jsx(_components.span, { className: "mord cjk_fallback", children: "域指示结点的后继" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.19em" }, children: _jsx(_components.span, {}) }) })] }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "2.9em" }, children: _jsx(_components.span, {}) }) })] }) }) })] }), _jsx(_components.span, { className: "tag", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "3.4em" }, children: [_jsxs(_components.span, { style: { top: "-5.4em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.75em" } }), _jsx(_components.span, { className: "eqn-num" })] }), _jsxs(_components.span, { style: { top: "-2.1em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.75em" } }), _jsx(_components.span, { className: "eqn-num" })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "2.9em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) }), "\n", _jsx(_components.p, { children: "线索二叉树的存储结构描述如下：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "typedef" }), " ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token class-name", children: "ThreadNode" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ElemType data", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 元素数据" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token class-name", children: "ThreadNode" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), "lchild", _jsx(_components.span, { className: "token punctuation", children: "," }), _jsx(_components.span, { className: "token operator", children: "*" }), "rchild", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 左、右孩子指针" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "int" }), " ltag", _jsx(_components.span, { className: "token punctuation", children: "," }), "rtag", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 左、右线索指针" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "ThreadNode", _jsx(_components.span, { className: "token punctuation", children: "," }), _jsx(_components.span, { className: "token operator", children: "*" }), "ThreadTree", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] })] }) }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "线索链表" }), "：以这种结点结构构成的二叉链表作为二叉树的存储结构，称为", _jsx(_components.em, { children: "线索链表" }), "。"] }), "\n", _jsxs(_components.li, { children: ["其中指向结点前驱和后继的指针称为", _jsx(_components.em, { children: "线索" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "线索二叉树" }), "：加上线索的二叉树称为", _jsx(_components.em, { children: "线索二叉树" }), "。"] }), "\n"] }), "\n", _jsxs(_components.h4, { id: "2-中序线索二叉树的构造", children: [_jsx(_components.a, { className: "heading-link", href: "#2-中序线索二叉树的构造", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "2. 中序线索二叉树的构造"] }), "\n", _jsx(_components.p, { children: "二叉树的线索化是将二叉链表中的空指针改为指向前驱或后继的线索。而前驱或后继的信息只有在遍历时才能得到，因此线索化的实质就是遍历一次二叉树。" }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "tip", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "命题追踪" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "中序线索二叉树中线索的指向" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["以中序线索二叉树的建立为例。附设指针", _jsx(_components.em, { children: "pre" }), "指向刚刚访问过的结点，指针", _jsx(_components.em, { children: "p" }), "指向正在访问的结点，即", _jsx(_components.em, { children: "pre" }), "指向", _jsx(_components.em, { children: "p" }), "的前驱。"] }), "\n", _jsxs(_components.p, { children: ["在中序遍历的过程中，检查", _jsx(_components.em, { children: "p" }), "的左指针是否为空，若为空就将它指", _jsx(_components.em, { children: "pre" }), "；检查", _jsx(_components.em, { children: "pre" }), "的右指针是否为空，若为空就将它指向", _jsx(_components.em, { children: "p" }), "，如图5.17所示。"] }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512032225552.png", alt: "image.png", className: "internal" }), "\n", _jsx(_components.p, { children: "通过中序遍历对二叉树线索化的递归算法如下：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "InThread" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "ThreadTree ", _jsx(_components.span, { className: "token operator", children: "&" }), "p", _jsx(_components.span, { className: "token punctuation", children: "," }), "ThreadTree ", _jsx(_components.span, { className: "token operator", children: "&" }), "pre", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "!=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "InThread" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild", _jsx(_components.span, { className: "token punctuation", children: "," }), "pre", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 递归，线索化左子树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild", _jsx(_components.span, { className: "token operator", children: "==" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), " ", _jsx(_components.span, { className: "token comment", children: "// 当前结点的左子树为空" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" p", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild", _jsx(_components.span, { className: "token operator", children: "=" }), "pre", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 建立当前结点的前驱线索" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" p", _jsx(_components.span, { className: "token operator", children: "->" }), "ltag", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "pre", _jsx(_components.span, { className: "token operator", children: "!=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token operator", children: "&&" }), "pre", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token operator", children: "==" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), " ", _jsx(_components.span, { className: "token comment", children: "// 前驱结点非空且其右子树为空" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" pre", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token operator", children: "=" }), "p", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 建立前驱结点的后继线索" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" pre", _jsx(_components.span, { className: "token operator", children: "->" }), "rtag", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" pre", _jsx(_components.span, { className: "token operator", children: "=" }), "p", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 标记当前结点称为刚刚访问过的结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "InThread" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token punctuation", children: "," }), "pre", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 递归，线索化右子树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "值得注意的是，这里的代码是C++风格的，在C中是没有引用传值这样的语法的，所以该实现放在C中会报错。鉴于某些学校的机试是纯C的，最好的方法是按照C语言来准备：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["ThreadTree pre ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "// C语言版本 - 使用双重指针传递pre" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "InThread" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "ThreadTree p", _jsx(_components.span, { className: "token punctuation", children: "," }), " ThreadTree ", _jsx(_components.span, { className: "token operator", children: "*" }), "pre", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p ", _jsx(_components.span, { className: "token operator", children: "!=" }), " ", _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "InThread" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild", _jsx(_components.span, { className: "token punctuation", children: "," }), " pre", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 递归，线索化左子树" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 处理左线索" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild ", _jsx(_components.span, { className: "token operator", children: "==" }), " ", _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), " ", _jsx(_components.span, { className: "token comment", children: "// 当前结点的左子树为空" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" p", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), "pre", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 建立当前结点的前驱线索" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" p", _jsx(_components.span, { className: "token operator", children: "->" }), "ltag ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "// 处理右线索（注意先检查pre是否为NULL）" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token operator", children: "*" }), "pre ", _jsx(_components.span, { className: "token operator", children: "!=" }), " ", _jsx(_components.span, { className: "token constant", children: "NULL" }), " ", _jsx(_components.span, { className: "token operator", children: "&&" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token operator", children: "*" }), "pre", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token operator", children: "->" }), "rchild ", _jsx(_components.span, { className: "token operator", children: "==" }), " ", _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), " \n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token operator", children: "*" }), "pre", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token operator", children: "->" }), "rchild ", _jsx(_components.span, { className: "token operator", children: "=" }), " p", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 建立前驱结点的后继线索" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token operator", children: "*" }), "pre", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token operator", children: "->" }), "rtag ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token operator", children: "*" }), "pre ", _jsx(_components.span, { className: "token operator", children: "=" }), " p", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 标记当前结点称为刚刚访问过的结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "InThread" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token punctuation", children: "," }), " pre", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 递归，线索化右子树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "abstract", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "解释" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsxs(_components.p, { children: ["下面对该代码作出解释，", _jsx(_components.code, { className: "language-auto", children: "pre" }), " 是一个指针，但在函数中是一个指向 ", _jsx(_components.code, { className: "language-auto", children: "pre" }), " 指针的指针。这么做是因为 ", _jsx(_components.code, { className: "language-auto", children: "pre" }), " 是一个全局指针，但我们需要在函数中去修改 ", _jsx(_components.code, { className: "language-auto", children: "pre" }), " 指针所指向的结点。如果指针修改 ", _jsx(_components.code, { className: "language-auto", children: "p" }), " 指针指向结点的结构体中的属性，就没必要这样，但这里需要修改指针本身，就需要用双指针了。我们知道，函数中修改行参只会修改函数中俄的副本，并不会修改函数外的参数本身。"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "通过中序遍历建立中序线索二叉树的主过程算法如下：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "CreateInThread" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "ThreadTree T", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ThreadTree pre", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token operator", children: "!=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), " ", _jsx(_components.span, { className: "token comment", children: "// 非空二叉树，线索化" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "InThread" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token punctuation", children: "," }), "pre", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 线索化二叉树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" pre", _jsx(_components.span, { className: "token operator", children: "->" }), "child", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 处理遍历的最后一个结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" pre", _jsx(_components.span, { className: "token operator", children: "->" }), "rtag", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["为方便起见，可在二叉树的线索链表上也添加一个头结点，令其", _jsx(_components.em, { children: "lchild" }), "域的指针指向二叉树的根结点，其", _jsx(_components.em, { children: "rchild" }), "域的指针指向中序遍历时访问的最后一个结点；"] }), "\n", _jsxs(_components.li, { children: ["令二叉树中序序列中的第一个结点", _jsx(_components.em, { children: "lchild" }), "域指针和最后一个结点的", _jsx(_components.em, { children: "rchild" }), "域指针均指向头结点。"] }), "\n"] }), "\n", _jsx(_components.p, { children: "这好比为二叉树建立了一个双向线索链表，方便从前往后或从后往前对线索二叉树进行遍历，如图5.18所示。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512041900888.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.h4, { id: "3-中序线索二叉树的遍历", children: [_jsx(_components.a, { className: "heading-link", href: "#3-中序线索二叉树的遍历", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "3. 中序线索二叉树的遍历"] }), "\n", _jsx(_components.p, { children: "中序线索二叉树的结点中隐含了线索二叉树的前驱和后继信息。" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "在对其进行遍历时，只要先找到序列中的第一个结点，然后依次找到结点的后继，直至其后继为空。" }), "\n", _jsxs(_components.li, { children: ["在中序线索二叉树中找到结点后继的规律是：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "若其右标志为“1”，则右链为线索，指示其后继，" }), "\n", _jsx(_components.li, { children: "否则遍历右子树中第一个访问的结点（右子树中最左下的结点）为后继。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "求中序线索二叉树的中序序列下的第一个结点：" }), "\n"] }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["ThreadNode ", _jsx(_components.span, { className: "token operator", children: "*" }), _jsx(_components.span, { className: "token function", children: "Firstnode" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "ThreadNode ", _jsx(_components.span, { className: "token operator", children: "*" }), "p", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "while" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "ltag", _jsx(_components.span, { className: "token operator", children: "==" }), _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " p", _jsx(_components.span, { className: "token operator", children: "=" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "lchild", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 最左下结点（不一定是叶子结点）" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " p", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.ol, { start: "2", children: ["\n", _jsxs(_components.li, { children: ["求中序线索二叉树中结", _jsx(_components.em, { children: "p" }), "在中序序列下的后继："] }), "\n"] }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: ["ThreadNode ", _jsx(_components.span, { className: "token operator", children: "*" }), _jsx(_components.span, { className: "token function", children: "Nextnode" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "ThreadNode ", _jsx(_components.span, { className: "token operator", children: "*" }), "p", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "rtag", _jsx(_components.span, { className: "token operator", children: "==" }), _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token keyword", children: "return" }), " ", _jsx(_components.span, { className: "token function", children: "Firstnode" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 右子树中最左下结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "else" }), " ", _jsx(_components.span, { className: "token keyword", children: "return" }), " p", _jsx(_components.span, { className: "token operator", children: "->" }), "rchild", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 若rtag==1则直接返回后继线索" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["这里 ", _jsx(_components.code, { className: "language-auto", children: "p->rtag==0" }), " 说明没有线索，直接往后找中序遍历的下一个结点。"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["显然最后一个结点也是同样的原理，只需要把 ", _jsx(_components.code, { className: "language-auto", children: "lchild" }), " 改成 ", _jsx(_components.code, { className: "language-auto", children: "r" }), " 就好了，前驱的算法就是左子树的最右下角结点，两种方法完全对称。"] }), "\n", _jsxs(_components.ol, { start: "3", children: ["\n", _jsx(_components.li, { children: "利用上面两个算法，可写出不含头结点的中序线索二叉树的中序遍历的算法：" }), "\n"] }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "Inorder" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "ThreadNode ", _jsx(_components.span, { className: "token operator", children: "*" }), "T", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "ThreadNode ", _jsx(_components.span, { className: "token operator", children: "*" }), "p", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token function", children: "Firstnode" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "T", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "p", _jsx(_components.span, { className: "token operator", children: "!=" }), _jsx(_components.span, { className: "token constant", children: "NULL" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " p", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token function", children: "Nextnode" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token function", children: "visit" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "p", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.h4, { id: "4-先序线索二叉树和后序线索二叉树", children: [_jsx(_components.a, { className: "heading-link", href: "#4-先序线索二叉树和后序线索二叉树", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "4. 先序线索二叉树和后序线索二叉树"] }), "\n", _jsxs(_components.p, { children: ["上面给出了建立中序线索二叉树的代码，建立先序线索二叉树和建立后序线索二叉树的代码类似，只需变动", _jsx(_components.strong, { children: "线索化改造的代码段" }), "与", _jsx(_components.strong, { children: "调用线索化左右子树递归函数的位置" }), "。"] }), "\n", _jsxs(_components.h5, { id: "1先序线索二叉树", children: [_jsx(_components.a, { className: "heading-link", href: "#1先序线索二叉树", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "（1）先序线索二叉树"] }), "\n", _jsxs(_components.p, { children: ["以图5.19（a）的二叉树为例给出手工求", _jsx(_components.strong, { children: "先序线索二叉树" }), "的过程："] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["先序序列为", _jsx(_components.em, { children: "ABCDF" }), "；"] }), "\n", _jsx(_components.li, { children: "然后依次判断每个结点的左右链域，若为空，则将其改造为线索。" }), "\n", _jsxs(_components.li, { children: ["结点", _jsx(_components.em, { children: "A，B" }), "均有左右孩子；"] }), "\n", _jsxs(_components.li, { children: ["结点", _jsx(_components.em, { children: "C" }), "无左孩子，将左链域指向前驱", _jsx(_components.em, { children: "B" }), "，无右孩子，将右链域指向后继", _jsx(_components.em, { children: "D" }), "；"] }), "\n", _jsxs(_components.li, { children: ["结点", _jsx(_components.em, { children: "D" }), "无左孩子，将左链域指向前驱", _jsx(_components.em, { children: "C" }), "，无右孩子，将右链域指向后继", _jsx(_components.em, { children: "F" }), "；"] }), "\n", _jsxs(_components.li, { children: ["结点", _jsx(_components.em, { children: "F" }), "无左孩子，将左链域指向前驱", _jsx(_components.em, { children: "D" }), "，无右孩子，也无后继，所以置空；"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "得到的先序线索二叉树如图5.18（b）所示。" }), "\n"] }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512042224846.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.h5, { id: "2后序线索二叉树", children: [_jsx(_components.a, { className: "heading-link", href: "#2后序线索二叉树", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "（2）后序线索二叉树"] }), "\n", _jsx(_components.p, { children: "求后序线索二叉树的过程：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["后序序列为", _jsx(_components.em, { children: "CDBFA" }), "；"] }), "\n", _jsxs(_components.li, { children: ["结点", _jsx(_components.em, { children: "C" }), "无左孩子，也无前驱，所以置空，无右孩子，将右链域指向后继", _jsx(_components.em, { children: "D" }), "；"] }), "\n", _jsxs(_components.li, { children: ["结点", _jsx(_components.em, { children: "D" }), "无左孩子，将左链域指向前驱", _jsx(_components.em, { children: "C" }), "，无右孩子，将右链域指向后继", _jsx(_components.em, { children: "B" }), "；"] }), "\n", _jsxs(_components.li, { children: ["结点", _jsx(_components.em, { children: "F" }), "无左孩子，将左链域指向前驱", _jsx(_components.em, { children: "B" }), "，无右孩子，将右链域指向后继", _jsx(_components.em, { children: "A" }), "；"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "纯粹的废话。" }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "得到的线索二叉树如图5.19（c）所示。" }), "\n"] }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512042224837.png", alt: "image.png", className: "internal" }), "\n", _jsx(_components.p, { children: "如何在先序线索二叉树中找结点的后继？" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "若有左孩子，则左孩子就是其后继；" }), "\n", _jsx(_components.li, { children: "若无左孩子但有右孩子，则右孩子就是其后继；" }), "\n", _jsx(_components.li, { children: "若为叶结点，则右链域直接指示了结点的后继。" }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "tip", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "命题追踪" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "后序线索二叉树中线索的指向" }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "在后序线索二叉树中找结点的后继较为复杂，可分为三种情况：" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["若结点 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "x" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "x" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] }) })] }), " 是二叉树的根，则其后继为空；"] }), "\n", _jsxs(_components.li, { children: ["若结点 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "x" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "x" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] }) })] }), " 是其双亲的右孩子，或是其双亲的左孩子且其双亲没有右子树，则其后继即双亲；"] }), "\n", _jsxs(_components.li, { children: ["若结点 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "x" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "x" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] }) })] }), " 是其双亲的左孩子，且其双亲有右子树，则其后继为双亲的右子树上按后序遍历列出的第一个结点。"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["图5.19（c）中找结点", _jsx(_components.em, { children: "B" }), "的后继无法通过链域找到，可见在后序线索二叉树上找后继时需知道结点双亲，即需采用带标志域的三叉链作为存储结构。"] }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512042235976.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.h2, { id: "54-树森林", children: [_jsx(_components.a, { className: "heading-link", href: "#54-树森林", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5.4 树、森林"] }), "\n", _jsx(_components.p, { children: "树的存储方式有多种，既可采用顺序存储结构，又可采用链式存储结构，但无论采用何种存储方式，都要求能唯一地反映各节点之间的逻辑关系，这里介绍3中常用的存储方式。" }), "\n", _jsxs(_components.h3, { id: "1-双亲表示法", children: [_jsx(_components.a, { className: "heading-link", href: "#1-双亲表示法", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "1. 双亲表示法"] }), "\n", _jsx(_components.p, { children: "这种存储结构采用一组连续空间来存储每个节点，同时在每个结点中增设一个伪指针，指示其双亲节点在数组中的位置。如图5.20所示，根结点下标为0，其伪指针域为-1。" }), "\n", _jsx(_components.p, { children: "双亲表示法的存储结构描述如下：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsxs(_components.span, { className: "token macro property", children: [_jsx(_components.span, { className: "token directive-hash", children: "#" }), _jsx(_components.span, { className: "token directive keyword", children: "define" }), " ", _jsx(_components.span, { className: "token macro-name", children: "MAX_TREE_SIZE" }), " ", _jsx(_components.span, { className: "token expression", children: _jsx(_components.span, { className: "token number", children: "100" }) })] }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "typedef" }), " ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ElemType data", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "int" }), " parent", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), " PTNode", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "typedef" }), " ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" PTNode nodes", _jsx(_components.span, { className: "token punctuation", children: "[" }), "MAS_TREE_SIZE", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "int" }), " n", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), " PTree", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] })] }) }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512101905513.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "每个结点都知道自己的双亲是谁，这样也能唯一表示一棵树了。" }), "\n"] }), "\n", _jsx(_components.p, { children: "双亲表示法利用了每个结点（根结点除外）只有唯一的双亲的性质，可以很快地得到每个结点的双亲结点，但求结点的孩子时需要遍历整个结构。" }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "warning", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "注意" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "区别树的顺序存储结构与二叉树的顺序存储结构：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "在树的顺序存储结构中，数组下标代表结点的编号，下标中所存的内容指示了结点之间的关系。" }), "\n", _jsx(_components.li, { children: "而在二叉树的顺序存储结构中，数组下标既代表了结点的编号，又指示了二叉树中各结点之间的关系。" }), "\n", _jsx(_components.li, { children: "当然，二叉树属于树，因此二叉树也可用树的存储结构来存储，但树却不都能用二叉树的存储结构来存储。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "这个注意的本质是因为二叉树中存在结点个数的信息，这个信息是很强烈的，但并不是所有的树的都只有两个结点。广义上的树的结点树是未知的，因此不能通过下标来直接找到孩子，必须通过存储位置的方式来保存位置信息。" }), "\n"] }), "\n", _jsxs(_components.h3, { id: "2-孩子表示法", children: [_jsx(_components.a, { className: "heading-link", href: "#2-孩子表示法", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "2. 孩子表示法"] }), "\n", _jsxs(_components.p, { children: ["孩子表示法是将每个结点的孩子结点视为一个线性表，且以单链表作为存储结构，则 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个结点就有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个孩子链表（叶节点的孩子链表为空表）。而 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个头指针又组成一个线性表，为便于查找，可采用顺序存储结构。图5.21(a)是图5.20(a)中的树的孩子表示法。"] }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512101919545.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.p, { children: ["与双亲表示法相反，孩子表示法寻找孩子的操作非常方便，而寻找双亲的操作则需要遍历 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个结点中孩子链表指针域所指向的 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个孩子链表。"] }), "\n", _jsxs(_components.h3, { id: "3-孩子兄弟表示法", children: [_jsx(_components.a, { className: "heading-link", href: "#3-孩子兄弟表示法", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "3. 孩子兄弟表示法"] }), "\n", _jsxs(_components.p, { children: ["孩子兄弟表示法也称", _jsx(_components.em, { children: "二叉树表示法" }), "，即以二叉链表作为树的存储结构。孩子兄弟表示法使每个结点包括三部分内容：结点值、指向结点第一个孩子结点的指针，以及指向结点下一个兄弟结点的指针（沿此域可以找到结点的所有兄弟结点），如图5.21(b)所示。"] }), "\n", _jsx(_components.p, { children: "孩子兄弟表示法的存储结构描述如下：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "typedef" }), " ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token class-name", children: "CSNode" }), " ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ElemType data", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 数据域" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "struct" }), " ", _jsx(_components.span, { className: "token class-name", children: "CSNode" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), "firstchild", _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), "nextsibling", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 第一个孩子和右兄弟指针" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), " CSNode", _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), "CSTree", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] })] }) }), "\n", _jsxs(_components.p, { children: ["孩子兄弟表示法比较灵活，其最大的优点是可以方便地实现树转换为二叉树的操作，易于查找结点的孩子等，但缺点是从当前结点查找其双亲结点比较麻烦。若为每个结点增设一个 ", _jsx(_components.code, { className: "language-auto", children: "parent" }), " 域指向其父结点，则查找结点的父结点也很方便。"] }), "\n", _jsxs(_components.h3, { id: "542-树森林与二叉树的转换", children: [_jsx(_components.a, { className: "heading-link", href: "#542-树森林与二叉树的转换", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5.4.2 树、森林与二叉树的转换"] }), "\n", _jsx(_components.p, { children: "二叉树和树都可以用二叉链表作为存储结构。从物理结构上来看，树的孩子兄弟表示法与二叉树的二叉链表表示法是相同的，因此可以用同一存储结构的不同解释将一棵树转换为二叉树。" }), "\n", _jsxs(_components.h4, { id: "1-树转换为二叉树", children: [_jsx(_components.a, { className: "heading-link", href: "#1-树转换为二叉树", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "1. 树转换为二叉树"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "info", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "命题追踪" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "树和二叉树的转换及相关性质的推理" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["树转换为二叉树的规则，每个结点的左指针指向它的第一个孩子，右指针指向它在树中的相邻右兄弟，这个规则也称", _jsx(_components.em, { children: "左孩子右兄弟" }), "（就是前面说的孩子兄弟表示法）。"] }), "\n", _jsx(_components.p, { children: "根结点没有兄弟，因此树转换得到的二叉树没有右子树，如图5.22所示。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512102000400.png", alt: "image.png", className: "internal" }), "\n", _jsx(_components.p, { children: "树转换为二叉树的画法：" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "在兄弟结点之间加一连线" }), "\n", _jsx(_components.li, { children: "对每个结点，只保留它与第一个孩子的连线，而与其他孩子的连线全部抹掉；" }), "\n", _jsxs(_components.li, { children: ["以树为轴心，顺时针旋转 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "45" }), _jsx(_components.mi, { mathvariant: "normal", children: "°" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "45°" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord", children: "45°" })] }) })] }), "。"] }), "\n"] }), "\n", _jsxs(_components.h4, { id: "2-森林转换为二叉树", children: [_jsx(_components.a, { className: "heading-link", href: "#2-森林转换为二叉树", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "2. 森林转换为二叉树"] }), "\n", _jsx(_components.p, { children: "将森林转换为二叉树的规则与树类似。先将森林中的每棵树转换为二叉树，由于任意一棵树对应的二叉树的右子树必空，森林中各棵树的根也可视为兄弟关系，将第二棵树对应的二叉树当作一颗二叉树根的右子树…以此类推，就可以将森林转换为二叉树。" }), "\n", _jsx(_components.p, { children: "森林转换为二叉树的画法：" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "将森林中的每棵树转换成相应的二叉树；" }), "\n", _jsx(_components.li, { children: "每棵树的根也可视为兄弟关系，在每棵树的根之间加一根线；" }), "\n", _jsx(_components.li, { children: "以第一棵树的根为轴心顺时针旋转45°。\n或者先在森林中每棵树的根之间加一根连线，然后再采用树转换为二叉树的方法。" }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "从这段文字中可以看出，用图像（并用糖葫芦这样的话来描述）很好描述的东西，如果用文字写出来就显得抽象不少，可见文字的重要作用并不是让人读完后就能理解。相反，文字给学过的人看倒是尤其独特的意义——压缩信息，如果一个知识组块已经在你的脑子里，那么文字的作用不过是一个刺激这片区域的钥匙。但第一次去理解这段文字则是抽象的，及时我已经看懂了视频，会自己动手去转换二叉树也是如此。" }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "树转二叉树" }), "：在这里，普通的树就是一堆散落的结点，我们将其转换为二叉树就是将平行的结点作为“糖葫芦”串起来，放在父亲的右子树下。\n", _jsx(_components.strong, { children: "二叉树转树" }), "：而二叉树转树就是将这个糖葫芦拆开，从上往下（", _jsx(_components.strong, { children: "一定要层序遍历" }), "）依次把右子树这条“糖葫芦”拆开，拆开的结点放在其父结点下。"] }), "\n"] }), "\n", _jsxs(_components.h4, { id: "3-二叉树转换为森林", children: [_jsx(_components.a, { className: "heading-link", href: "#3-二叉树转换为森林", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "3. 二叉树转换为森林"] }), "\n", _jsx(_components.p, { children: "二叉树转换为森林的规则：若二叉树为空，则二叉树的根及其左子树为第一棵树的二叉树形式，所以将根的右链断开。二叉树根的右子树又可视为一个由除第一棵树外的森林转换后的二叉树，应用同样的方法，直到最后只剩一棵没有右子树的二叉树为止，最后将每棵二叉树依次转换为树，就得到了原森林（The Forest乱入hhh）。" }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "nmd这段文字真难读，不如把方法记住得了，没必要，试图理解这段文字是真的恶心。" }), "\n"] }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512112141395.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "逗你玩的。" }), "\n"] }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512112140521.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.h3, { id: "543-树和森林的遍历", children: [_jsx(_components.a, { className: "heading-link", href: "#543-树和森林的遍历", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5.4.3 树和森林的遍历"] }), "\n", _jsxs(_components.h4, { id: "1-树的遍历", children: [_jsx(_components.a, { className: "heading-link", href: "#1-树的遍历", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "1. 树的遍历"] }), "\n", _jsx(_components.p, { children: "树的遍历是指用某种方式访问树中的每个结点，且仅访问一次。主要有两种方式：" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "先根遍历" }), "。若树非空，则按如下规则遍历：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "先访问根结点。" }), "\n", _jsx(_components.li, { children: "再依次遍历根结点的每棵子树，遍历子树时仍遵循先根后子树的规则。\n其遍历顺序与这棵树相应的二叉树的先序序列相同。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "后根遍历" }), "。若树非空，则按如下规则遍历：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "先依次遍历根结点的每棵子树，遍历子树时仍遵循先子树后根的规则。" }), "\n", _jsx(_components.li, { children: "再访问根结点。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "其遍历序列与这棵树相应的二叉树的中序序列相同。" }), "\n", _jsxs(_components.p, { children: ["图5.22的树的先根遍历序列为", _jsx(_components.em, { children: "ABEFCDG" }), "，后根遍历序列为", _jsx(_components.em, { children: "EFBCGDA" }), "。\n另外，树也有层次遍历，与二叉树的层次遍历思想基本相同，即按照层序依次访问各结点。\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512112140521.png", alt: "image.png", className: "internal" })] }), "\n", _jsxs(_components.h4, { id: "2-森林的遍历", children: [_jsx(_components.a, { className: "heading-link", href: "#2-森林的遍历", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "2. 森林的遍历"] }), "\n", _jsx(_components.p, { children: "按照森林和树相互递归的定义，可得到森林的两种遍历方法。" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "先序遍历森林" }), "。若森林为非空，则按如下规则遍历：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "访问森林中第一棵树的根结点。" }), "\n", _jsx(_components.li, { children: "先序遍历除去第一棵树中根结点的子树森林。" }), "\n", _jsx(_components.li, { children: "先序遍历除去第一棵树之后剩余的树构成的森林。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "中序遍历森林" }), "。森林为非空时，按如下规则遍历：", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "中序遍历森林中第一棵树的根结点的子树森林。" }), "\n", _jsx(_components.li, { children: "访问第一棵树的根结点。" }), "\n", _jsx(_components.li, { children: "中序遍历除去第一棵树之后剩余的树构成的森林。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["图5.23的森林的先序遍历序列为", _jsx(_components.em, { children: "ABCDEFGHI" }), "，中序遍历为", _jsx(_components.em, { children: "BCDAFEHIG" }), "。"] }), "\n", _jsx(_components.p, { children: "当森林转换成二叉树时，其第一棵树的子树森林转换成左子树，剩余树的森林转换成右子树，可知森林的先序和中序遍历即其对应二叉树的先序和中序遍历。" }), "\n", _jsx(_components.p, { children: "树和森林的遍历与二叉树的遍历关系见5.1。" }), "\n", _jsx(_components.div, { className: "overflow-x-auto", children: _jsxs(_components.table, { children: [_jsx(_components.thead, { children: _jsxs(_components.tr, { children: [_jsx(_components.th, { children: "树" }), _jsx(_components.th, { children: "森林" }), _jsx(_components.th, { children: "二叉树" })] }) }), _jsxs(_components.tbody, { children: [_jsxs(_components.tr, { children: [_jsx(_components.td, { children: "先根遍历" }), _jsx(_components.td, { children: "先序遍历" }), _jsx(_components.td, { children: "先序遍历" })] }), _jsxs(_components.tr, { children: [_jsx(_components.td, { children: "后根遍历" }), _jsx(_components.td, { children: "中序遍历" }), _jsx(_components.td, { children: "中序遍历" })] })] })] }) }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "注意" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "森林的遍历方法的命名按照严蔚敏的经典教材。有些教材也将森林的中序遍历成为后序遍历，称中序遍历是相对其二叉树而言的，称后序遍历是因为根确实是最后才访问的。" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "55-树与二叉树的应用", children: [_jsx(_components.a, { className: "heading-link", href: "#55-树与二叉树的应用", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5.5 树与二叉树的应用"] }), "\n", _jsxs(_components.h3, { id: "551-哈夫曼树和哈夫曼编码", children: [_jsx(_components.a, { className: "heading-link", href: "#551-哈夫曼树和哈夫曼编码", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5.5.1 哈夫曼树和哈夫曼编码"] }), "\n", _jsxs(_components.h3, { id: "1-哈夫曼树的定义", children: [_jsx(_components.a, { className: "heading-link", href: "#1-哈夫曼树的定义", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "1. 哈夫曼树的定义"] }), "\n", _jsx(_components.p, { children: "在介绍哈夫曼树之前，先介绍几个相关的概念：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "权" }), "：在许多应用中，树中结点尝尝被赋予一个表示某种意义的数值，称为该节点的", _jsx(_components.em, { children: "权" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "带权路径长度" }), "：", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "结点的带权路径长度" }), "：从树的根到一个结点的长度与该节点上权值的乘积，称为该", _jsx(_components.em, { children: "节点的" }), _jsx(_components.strong, { children: "带权路径长度" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "树的带权路径长度" }), "：树中所有", _jsx(_components.strong, { children: "叶节点" }), "的带权路径长度之和称为该", _jsx(_components.em, { children: "树的" }), _jsx(_components.strong, { children: "带权路径长度" }), "，记为：", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "W" }), _jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "L" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "w" }), _jsx(_components.mi, { children: "i" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "l" }), _jsx(_components.mi, { children: "i" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "WPL=\\sum_{i=1}^{n} w_{i}l_{i}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "W" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.104em", verticalAlign: "-0.2997em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0269em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0269em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0197em" }, children: "l" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0197em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] })] }), "\n", _jsxs(_components.li, { children: ["式中，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "w" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "w_{i}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0269em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0269em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " 是第 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }), " 个结点所带的权值，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "l" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "l_{i}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8444em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0197em" }, children: "l" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0197em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " 是该", _jsx(_components.strong, { children: "叶结点" }), "到根结点的路径长度。"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "哈夫曼树" }), "：在含有 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个带权叶结点的二叉树中，其中带权路径长度（", _jsx(_components.em, { children: "WPL" }), "）最小的二叉树成为", _jsx(_components.strong, { children: "哈夫曼树" }), "，也叫做", _jsx(_components.strong, { children: "最优二叉树" }), "。"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: ["例如，图5.24中的3棵二叉树都有4个叶结点 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "b" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "d" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a,b,c,d" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "d" })] }) })] }), "，分别带权 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "7" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "5" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "4" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "7,5,2,4" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord", children: "7" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "5" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "4" })] }) })] }), "，它们的带权路径长度分别为：", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512141821173.png", alt: "image.png", className: "internal" })] }), "\n"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { children: "W" }), _jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "L" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "7" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "5" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "4" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "36" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(a)WPL=7\\times2+5\\times 2+ 2 \\times 2 + 4 \\times 2 = 36" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "W" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "7" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "5" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "4" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "36" })] })] })] }) }), "\n", _jsx(_components.li, { children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "b" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { children: "W" }), _jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "L" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "4" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "7" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "5" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "46" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(b)WPL=4\\times 2+7 \\times 3 + 5 \\times 3 + 2 \\times 1 = 46" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "W" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "4" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "7" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "5" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "46" })] })] })] }) }), "\n", _jsx(_components.li, { children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { children: "W" }), _jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "L" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "7" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "5" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "4" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "35" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(c)WPL = 7 \\times 1 + 5 \\times 2 + 2 \\times 3 + 4 \\times 3= 35" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "W" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "7" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "5" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "4" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "35" })] })] })] }) }), "\n"] }), "\n", _jsxs(_components.p, { children: ["其中，图5.24（c）树的 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "W" }), _jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "L" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "WPL" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "W" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] }) })] }), " 最小。可以验证，它恰好为哈夫曼树。"] }), "\n", _jsxs(_components.h3, { id: "2-哈夫曼树的构造", children: [_jsx(_components.a, { className: "heading-link", href: "#2-哈夫曼树的构造", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "2. 哈夫曼树的构造"] }), "\n", _jsxs(_components.p, { children: ["给定 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个权值分别为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "w" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "w" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mo, { children: "…" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "w" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "w_{1},w_{2},\\dots,w_{n}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0269em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0269em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0269em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0269em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "minner", children: "…" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0269em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0269em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " 的结点，构造哈夫曼树的算法描述如下："] }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["将这 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个结点分别作为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 棵仅含一个结点的二叉树，构成森林 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "F" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "F" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "F" })] }) })] }), "。"] }), "\n", _jsxs(_components.li, { children: ["构造一个新结点，从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "F" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "F" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "F" })] }) })] }), " 中选取两棵根结点权值最小的树作为新结点的左、右子树，并且将新结点的权值置为左、右子树上根结点的权值之和。"] }), "\n", _jsxs(_components.li, { children: ["从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "F" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "F" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "F" })] }) })] }), " 中删除刚才选出的两棵树，同时将新得到的树加入 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "F" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "F" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "F" })] }) })] }), " 中。"] }), "\n", _jsxs(_components.li, { children: ["重复步骤 ", _jsx(_components.code, { className: "language-auto", children: "2." }), " 和 ", _jsx(_components.code, { className: "language-auto", children: "3." }), "，直至F ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "F" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "F" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "F" })] }) })] }), " 中只剩下一棵树为止。"] }), "\n"] }), "\n", _jsx(_components.p, { children: "从上述构造可以看出哈夫曼树具有如下特点：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "每个初始结点最终都成为叶结点，且权值越小的结点到根结点的路径长度越大。" }), "\n", _jsxs(_components.li, { children: ["构造过程中共新建了 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n-1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), " 个结点（双分支结点），因此哈夫曼树的结点总数为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2n - 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "。"] }), "\n", _jsx(_components.li, { children: "每次构造都选择2棵树作为新结点的孩子，因此哈夫曼树红不存在度为1的结点。" }), "\n"] }), "\n", _jsxs(_components.p, { children: ["例如，权值 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "{" }), _jsx(_components.mn, { children: "7" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "5" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "4" }), _jsx(_components.mo, { stretchy: "false", children: "}" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\{ 7,5,2,4 \\}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "{" }), _jsx(_components.span, { className: "mord", children: "7" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "5" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "4" }), _jsx(_components.span, { className: "mclose", children: "}" })] }) })] }), " 的哈夫曼树的构造过程如图5.25所示。"] }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512141842393.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.h3, { id: "3-哈夫曼编码", children: [_jsx(_components.a, { className: "heading-link", href: "#3-哈夫曼编码", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "3. 哈夫曼编码"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "固定长度编码" }), "：在数据通信中，若对每个字符用相等长度的二进制位表示，称这种编码方式为", _jsx(_components.strong, { children: "固定长度编码" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "可变长度编码" }), "：若允许对不同字符用不等长的二进制位表示，则这种编码方式称为", _jsx(_components.strong, { children: "可变长度编码" }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "前缀编码" }), "：若没有一个编码是另一个编码的前缀，则称这样的编码为", _jsx(_components.strong, { children: "前缀编码" }), "。", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "举例" }), "：设计字符串 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "A" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "B" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A,B" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0502em" }, children: "B" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "C" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0715em" }, children: "C" })] }) })] }), " 对应的编码 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "10" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "0,10" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "10" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "110" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "110" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "110" })] }) })] }), " 是前缀编码。对前缀编码的解码很简单，因为没有一个编码是其他编码的前缀。所以识别出第一个编码，将它翻译为", _jsx(_components.em, { children: "原字符" }), "，再对剩余的码串执行同样的解码操作。"] }), "\n", _jsxs(_components.li, { children: ["例如，码串 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "0010110" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "0010110" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0010110" })] }) })] }), " 可被唯一地翻译为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "A" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "A" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "B" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A,A,B" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0502em" }, children: "B" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "C" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0715em" }, children: "C" })] }) })] }), "。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "另举反例" }), "：若再将字符 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "D" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "D" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "D" })] }) })] }), " 的编码设计为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "11" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "11" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "11" })] }) })] }), "，此时 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "11" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "11" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "11" })] }) })] }), " 是 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "110" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "110" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "110" })] }) })] }), " 的前缀，则上述码串中的后三位就无法唯一翻译。"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsx(_components.p, { children: "可变长度编码比固定长度编码要好得多，其特点是对频率高的字符赋以短编码，而对频率较低的字符则赋以较长一些的编码，从而可以使字符的平均编码长度减短，起到压缩数据的效果。" }), "\n"] }), "\n", _jsx(_components.p, { children: "可以利用二叉树来设计二进制前缀编码。假设为A，B，C，D四个字符设计前缀编码，可以用图5.26所示的二叉树来表示，4个叶结点分别表示4个字符，且约定左分支表示0，右分支表示1，从根到叶结点的路径上用分支标记组成的序列作为该叶结点字符的编码，可以证明如此得到的必为前缀编码。由图5.26得到字符A，B，C，D的前缀编码分别为0，10，110，111。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512141855633.png", alt: "image.png", className: "internal" }), "\n", _jsx(_components.p, { children: "哈夫曼编码是一种非常有效的数据压缩编码。由哈夫曼树得到哈夫曼编码是很自然的过程。" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "首先，将每个字符当作一个独立的结点，其权值为它出现的频度（或次数），构造出对应的哈夫曼树。" }), "\n", _jsx(_components.li, { children: "然后，将从根到叶结点的路径上分支标记的字符串作为该字符的编码。" }), "\n"] }), "\n", _jsx(_components.p, { children: "图5.27所示为一个由哈夫曼树构造哈夫曼编码的示例，矩形方块表示字符及其出现的次数。" }), "\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512141901331.png", alt: "image.png", className: "internal" }), "\n", _jsxs(_components.p, { children: ["这棵哈夫曼树的 ", _jsx(_components.em, { children: "WPL" }), " 为："] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "W" }), _jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "L" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "45" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "13" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "12" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "16" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "4" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "5" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "9" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "224" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "WPL=1 \\times 45 + 3 \\times (13 + 12 + 16) + 4 \\times (5 + 9) = 224" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "W" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "45" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "13" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "12" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "16" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "4" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "5" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "9" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "224" })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["此处的 ", _jsx(_components.em, { children: "WPL" }), " 可视为最终编码得到的二进制编码的长度，共224位。若采用3位固定长度编码，则得到的二进制编码长度为300位，因此哈夫曼编码共压缩了25%的数据。利用哈夫曼树可以设计出总长度最短的二进制前缀编码。"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "note", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "注意" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, {}), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "左分支和右分支究竟是表示0还是表示1没有明确规定，因此构造出的哈夫曼树并不唯一；" }), "\n", _jsx(_components.li, { children: "此外，如有若干权值相同的结点，则构造出的哈夫曼树更能不同。" }), "\n", _jsx(_components.li, { children: "无论是上述两种情况种的哪一种，各哈夫曼树的带权路径长度WPL相同且为最优。" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "552-并查集", children: [_jsx(_components.a, { className: "heading-link", href: "#552-并查集", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "5.5.2 并查集"] }), "\n", _jsxs(_components.h4, { id: "1-并查集的概念", children: [_jsx(_components.a, { className: "heading-link", href: "#1-并查集的概念", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "1. 并查集的概念"] }), "\n", _jsx(_components.p, { children: "并查集是一种简单的集合表示，它支持以下3种操作：" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "Initial(S)" }), "：将结合 ", _jsx(_components.code, { className: "language-auto", children: "S" }), " 种的每个元素都初始化为只有一个单元素的子集合。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "Union(S,Root1，Root2)" }), "：把集合 ", _jsx(_components.code, { className: "language-auto", children: "S" }), " 种的子集合 ", _jsx(_components.code, { className: "language-auto", children: "Root2" }), " 并入子集合 ", _jsx(_components.code, { className: "language-auto", children: "Root1" }), "。要求 ", _jsx(_components.code, { className: "language-auto", children: "Root1" }), " 和 ", _jsx(_components.code, { className: "language-auto", children: "Root2" }), " 互不相交，否则不执行合并。"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.code, { className: "language-auto", children: "Find(S,x)" }), "：查找集合 ", _jsx(_components.code, { className: "language-auto", children: "S" }), " 种单元素 ", _jsx(_components.code, { className: "language-auto", children: "x" }), " 所在的子集合，并返回该子集合的根结点。"] }), "\n"] }), "\n", _jsxs(_components.h4, { id: "2-并查集的存储结构", children: [_jsx(_components.a, { className: "heading-link", href: "#2-并查集的存储结构", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "2. 并查集的存储结构"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["通常用树的", _jsx(_components.strong, { children: "双亲表示" }), "作为并查集的存储结构，每个", _jsx(_components.strong, { children: "子集合以一棵树表示" }), "。"] }), "\n", _jsxs(_components.li, { children: ["所有表示", _jsx(_components.strong, { children: "子集合的树" }), "，构成表示", _jsx(_components.strong, { children: "全集合的森林" }), "，存放在", _jsx(_components.strong, { children: "双亲表示数组" }), "内。"] }), "\n", _jsxs(_components.li, { children: ["通常用", _jsx(_components.strong, { children: "数组" }), "元素的", _jsx(_components.strong, { children: "下标表示元素名" }), "，用", _jsx(_components.strong, { children: "根结点" }), "的", _jsx(_components.strong, { children: "下标代表子集合名" }), "，根结点的双亲域为负数（可设置为该子集合元素数量的相反数）。"] }), "\n"] }), "\n", _jsxs(_components.div, { "data-callout": true, "data-callout-type": "abstract", children: ["\n", _jsx(_components.div, { "data-callout-title": true, children: "例子" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsxs(_components.p, { children: ["例如，若设有一个全集合为 ", _jsx(_components.code, { className: "language-auto", children: "S={0,1,2,3,4,5,6,7,8,9}" }), "，初始化时每个元素自成一个单元素子集合，每个子集合的数组值为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "-1" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" })] }) })] }), "，如图5.28所示："] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512141931883.png", alt: "image.png", className: "internal" }), "\n经过一段时间的计算后，这些子集合合并为3个更大的子集合，即 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { stretchy: "false", children: "{" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "6" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "7" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "8" }), _jsx(_components.mo, { stretchy: "false", children: "}" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { stretchy: "false", children: "{" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "4" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "9" }), _jsx(_components.mo, { stretchy: "false", children: "}" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mn, { children: "3" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { stretchy: "false", children: "{" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "3" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "5" }), _jsx(_components.mo, { stretchy: "false", children: "}" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_{1}=\\{ 0,6,7,8 \\},S_{2}=\\{ 1,4,9 \\},S_{3}=\\{ 2,3,5 \\}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0576em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "{" }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "6" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "7" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "8" }), _jsx(_components.span, { className: "mclose", children: "}" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0576em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "{" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "4" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "9" }), _jsx(_components.span, { className: "mclose", children: "}" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0576em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "{" }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "5" }), _jsx(_components.span, { className: "mclose", children: "}" })] })] })] }), "，此时并查集的树形和存储结构如图5.29所示：（这里图中的 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mn, { children: "3" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_{3}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0576em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " 树的根结点明显错了）\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512141935848.png", alt: "image.png", className: "internal" }), "\n为了得到两个子集合的并，只需将其中一个子集合根结点的双亲指针指向另一个集合的根结点。因此，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { children: "∪" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mn, { children: "2" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_{1} \\cup S_{2}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0576em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "∪" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0576em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] }), " 可以具有如图5.30所示的表示。\n", _jsx(_components.img, { src: "https://ccccooh.oss-cn-hangzhou.aliyuncs.com/img/202512141937114.png", alt: "image.png", className: "internal" })] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["在采用树的双亲指针数组表示作为并查集的存储表示时，集合元素的编号从 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "0" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "0" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] }) })] }), " 到 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "I" }), _jsx(_components.mi, { children: "Z" }), _jsx(_components.mi, { children: "E" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "SIZE-1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0576em" }, children: "S" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0785em" }, children: "I" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0715em" }, children: "Z" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0576em" }, children: "E" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "。其中 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "I" }), _jsx(_components.mi, { children: "Z" }), _jsx(_components.mi, { children: "E" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "SIZE" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0576em" }, children: "S" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0785em" }, children: "I" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0715em" }, children: "Z" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0576em" }, children: "E" })] }) })] }), " 是最大元素的个数。"] }), "\n", _jsxs(_components.h4, { id: "3-并查集的基本实现", children: [_jsx(_components.a, { className: "heading-link", href: "#3-并查集的基本实现", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "3. 并查集的基本实现"] }), "\n", _jsx(_components.p, { children: "并查集的结构定义如下：" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsxs(_components.span, { className: "token macro property", children: [_jsx(_components.span, { className: "token directive-hash", children: "#" }), _jsx(_components.span, { className: "token directive keyword", children: "define" }), " ", _jsx(_components.span, { className: "token macro-name", children: "SIZE" }), " ", _jsx(_components.span, { className: "token expression", children: _jsx(_components.span, { className: "token number", children: "100" }) })] }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "int" }), " UFSets", _jsx(_components.span, { className: "token punctuation", children: "[" }), "SIZE", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 集合元素数组(双亲指针数组)" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "下面是并查集主要运算的实现。" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "并查集的初始化操作" }), "\n"] }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "Initial" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), " S", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), " i", _jsx(_components.span, { className: "token operator", children: "=" }), _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "i", _jsx(_components.span, { className: "token operator", children: "<" }), "SIZE", _jsx(_components.span, { className: "token punctuation", children: ";" }), "i", _jsx(_components.span, { className: "token operator", children: "++" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// S就是并查集" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "i", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 每个自成单元素集合" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.ol, { start: "2", children: ["\n", _jsxs(_components.li, { children: ["并查集的 ", _jsx(_components.code, { className: "language-auto", children: "Find" }), " 操作"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["在并查集 ", _jsx(_components.code, { className: "language-auto", children: "S" }), " 中查找并返回包含元素 ", _jsx(_components.code, { className: "language-auto", children: "x" }), " 的树的根。"] }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "int" }), " ", _jsx(_components.span, { className: "token function", children: "Find" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), " S", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token keyword", children: "int" }), " x", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "while" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "x", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: ">=" }), " ", _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 循环寻找x的根" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" x", _jsx(_components.span, { className: "token operator", children: "=" }), "S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "x", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " x", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 根的S[]小于0" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "判断两个元素是否属于同一集合，只需分别找到它们的根，再比较根是否相同即可。" }), "\n", _jsxs(_components.ol, { start: "3", children: ["\n", _jsxs(_components.li, { children: ["并查集的 ", _jsx(_components.code, { className: "language-auto", children: "Union" }), " 操作"] }), "\n"] }), "\n", _jsx(_components.p, { children: "求两个不相交子集合的并集。若将两个元素所在的集合合并为一个集合，则需要先找到两个元素的根，再令一棵子集树的根指向另一棵子集树的根。" }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "Union" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), " S", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "," }), _jsx(_components.span, { className: "token keyword", children: "int" }), " Root1", _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token keyword", children: "int" }), " Root2", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "Root1", _jsx(_components.span, { className: "token operator", children: "==" }), "Root2", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token keyword", children: "return" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 要求Root1与Root2是不同的集合" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "Root2", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token operator", children: "=" }), "Root1", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 将根Root2连接到另一根Root1下面" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.p, { children: [_jsx(_components.code, { className: "language-auto", children: "Find" }), " 操作和 ", _jsx(_components.code, { className: "language-auto", children: "Union" }), " 操作的时间复杂度分别为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "d" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(d)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(1)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "，其中 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "d" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "d" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "d" })] }) })] }), " 为树的深度。"] }), "\n", _jsxs(_components.ol, { start: "4", children: ["\n", _jsx(_components.li, { children: "并查集实现的优化" }), "\n"] }), "\n", _jsxs(_components.p, { children: ["在极端情况下，", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 个元素构成的集合树的深度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), "，则 ", _jsx(_components.code, { className: "language-auto", children: "Find" }), " 操作的最坏时间复杂度为 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), "。改进的办法是：在做 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "U" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "o" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Union" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.109em" }, children: "U" }), _jsx(_components.span, { className: "mord mathnormal", children: "ni" }), _jsx(_components.span, { className: "mord mathnormal", children: "o" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " 操作之前，首先判别子集中的成员数量，然后令成员少的根指向成员多的根，即把小树合并到大树，为此可令根结点的绝对值保存集合树中的成员数量。"] }), "\n", _jsxs(_components.p, { children: ["（1）改进的 ", _jsx(_components.code, { className: "language-auto", children: "Union" }), " 操作"] }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "void" }), " ", _jsx(_components.span, { className: "token function", children: "Union" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), " S", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "," }), _jsx(_components.span, { className: "token keyword", children: "int" }), " Root1", _jsx(_components.span, { className: "token punctuation", children: "," }), _jsx(_components.span, { className: "token keyword", children: "int" }), " Root2", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "Root1", _jsx(_components.span, { className: "token operator", children: "==" }), "Root2", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token keyword", children: "return" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "if" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "Root2", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token operator", children: ">" }), "S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "Root1", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// （负数）Root2的结点数更少" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "Root1", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token operator", children: "+=" }), "S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "Root2", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 累加集合树的结点总数" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "Root2", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token operator", children: "=" }), "Root1", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 小树合并到大树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "else" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "{" }), " ", _jsx(_components.span, { className: "token comment", children: "// Root1的结点数更少" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "Root2", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token operator", children: "+=" }), "S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "Root1", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 累加结点总数" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "Root1", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token operator", children: "=" }), "Root2", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 小树合并到大树" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.p, { children: ["采用这种方法构造得到的集合树，其深度不超过 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "⌊" }), _jsxs(_components.msub, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "log" }), _jsx(_components.mo, { children: "⁡" })] }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "⌋" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lfloor \\log_{2}n+1 \\rfloor" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "⌊" }), _jsxs(_components.span, { className: "mop", children: [_jsxs(_components.span, { className: "mop", children: ["lo", _jsx(_components.span, { style: { marginRight: "0.0139em" }, children: "g" })] }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.207em" }, children: _jsxs(_components.span, { style: { top: "-2.4559em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2441em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "⌋" })] })] })] }), "。\n随着子集逐对合并，集合树的深度越来越大，为了进一步减少确定元素所在集合的时间，还可以进一步对上述 ", _jsx(_components.code, { className: "language-auto", children: "Find" }), " 操作进行优化，当所查元素 ", _jsx(_components.code, { className: "language-auto", children: "x" }), " 不在树的第二层时，在算法中增加一个", _jsx(_components.strong, { children: "压缩路径" }), "的功能，即将从根到元素 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "x" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "x" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] }) })] }), " 路径上的所有元素都变成根的孩子。"] }), "\n", _jsxs(_components.p, { children: ["（2）改进后的 ", _jsx(_components.code, { className: "language-auto", children: "Find" }), " 操作"] }), "\n", _jsx(_components.pre, { className: "language-c", children: _jsxs(_components.code, { className: "language-c code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "int" }), " ", _jsx(_components.span, { className: "token function", children: "Find" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token keyword", children: "int" }), " S", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token keyword", children: "int" }), " x", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "int" }), " root ", _jsx(_components.span, { className: "token operator", children: "=" }), " x", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "while" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "s", _jsx(_components.span, { className: "token punctuation", children: "[" }), "root", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: ">=" }), " ", _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 循环查找根" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" root ", _jsx(_components.span, { className: "token operator", children: "=" }), " S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "root", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "while" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "x ", _jsx(_components.span, { className: "token operator", children: "!=" }), " root", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "// 压缩路径" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "{" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "int" }), " t ", _jsx(_components.span, { className: "token operator", children: "=" }), " S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "x", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// t 指向 x 的父结点" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" S", _jsx(_components.span, { className: "token punctuation", children: "[" }), "x", _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " root", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// x 直接挂到根结点下面" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" x ", _jsx(_components.span, { className: "token operator", children: "=" }), " t", _jsx(_components.span, { className: "token punctuation", children: ";" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " root", _jsx(_components.span, { className: "token punctuation", children: ";" }), " ", _jsx(_components.span, { className: "token comment", children: "// 返回根结点编号" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token punctuation", children: "}" }), "\n"] })] }) }), "\n", _jsxs(_components.p, { children: ["通过 ", _jsx(_components.code, { className: "language-auto", children: "Find" }), " 操作的压缩路径优化后，可使集合树的深度不超过 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "O" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "α" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "O(\\alpha(n))" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "O" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: "))" })] }) })] }), "，其中 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "α" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\alpha(n)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " 是一个增长极其缓慢的函数，对于常见的正整数 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), "，通常 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "α" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "≤" }), _jsx(_components.mn, { children: "4" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\alpha(n)\\leq 4" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "4" })] })] })] }), "。\n并查集应用相关的举例见本书的配套课程。"] })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = { ..._provideComponents(), ...props.components }; return MDXLayout ? _jsx(MDXLayout, { ...props, children: _jsx(_createMdxContent, { ...props }) }) : _createMdxContent(props); } return { default: MDXContent };