{"id":27385,"date":"2025-04-22T14:44:15","date_gmt":"2025-04-22T07:44:15","guid":{"rendered":"https:\/\/interdata.vn\/blog\/?p=27385"},"modified":"2025-04-22T14:44:15","modified_gmt":"2025-04-22T07:44:15","slug":"cay-nhi-phan-la-gi","status":"publish","type":"post","link":"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/","title":{"rendered":"C\u00e2y Nh\u1ecb Ph\u00e2n L\u00e0 G\u00ec? (Binary Tree) &#8211; \u0110\u1eb7c \u0110i\u1ec3m &#038; V\u00ed D\u1ee5 D\u1ec5 Hi\u1ec3u"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">N\u1ed8I DUNG<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Cay-nhi-phan-Binary-Tree-la-gi\" >C\u00e2y nh\u1ecb ph\u00e2n (Binary Tree) l\u00e0 g\u00ec?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Tai-sao-goi-la-%E2%80%9CNhi-phan%E2%80%9D\" >T\u1ea1i sao g\u1ecdi l\u00e0 &#8220;Nh\u1ecb ph\u00e2n&#8221;?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Cac-thuat-ngu-quan-trong-can-biet-ve-Cay-Nhi-Phan\" >C\u00e1c thu\u1eadt ng\u1eef quan tr\u1ecdng c\u1ea7n bi\u1ebft v\u1ec1 C\u00e2y Nh\u1ecb Ph\u00e2n<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Nut-Node\" >N\u00fat (Node)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Canh-Edge\" >C\u1ea1nh (Edge)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Nut-Goc-Root-Node\" >N\u00fat G\u1ed1c (Root Node)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Nut-La-Leaf-Node\" >N\u00fat L\u00e1 (Leaf Node)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Nut-Cha-Parent-Node-Nut-Con-Child-Node\" >N\u00fat Cha (Parent Node) &amp; N\u00fat Con (Child Node)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Nut-Trung-Gian-Internal-Node\" >N\u00fat Trung Gian (Internal Node)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Cay-con-Subtree\" >C\u00e2y con (Subtree)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Do-sau-cua-Nut-Depth-of-a-Node\" >\u0110\u1ed9 s\u00e2u c\u1ee7a N\u00fat (Depth of a Node)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Chieu-cao-cua-Cay-Height-of-a-Tree\" >Chi\u1ec1u cao c\u1ee7a C\u00e2y (Height of a Tree)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Bac-cua-Nut-Degree-of-a-Node\" >B\u1eadc c\u1ee7a N\u00fat (Degree of a Node)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Dac-diem-va-tinh-chat-co-ban-cua-Cay-Nhi-Phan\" >\u0110\u1eb7c \u0111i\u1ec3m v\u00e0 t\u00ednh ch\u1ea5t c\u01a1 b\u1ea3n c\u1ee7a C\u00e2y Nh\u1ecb Ph\u00e2n<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Vi-du-minh-hoa-ve-Cay-Nhi-Phan-don-gian\" >V\u00ed d\u1ee5 minh h\u1ecda v\u1ec1 C\u00e2y Nh\u1ecb Ph\u00e2n \u0111\u01a1n gi\u1ea3n<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Gioi-thieu-mot-so-loai-Cay-Nhi-Phan-pho-bien\" >Gi\u1edbi thi\u1ec7u m\u1ed9t s\u1ed1 lo\u1ea1i C\u00e2y Nh\u1ecb Ph\u00e2n ph\u1ed5 bi\u1ebfn<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Cay-Nhi-Phan-Tim-Kiem-Binary-Search-Tree-%E2%80%93-BST\" >C\u00e2y Nh\u1ecb Ph\u00e2n T\u00ecm Ki\u1ebfm (Binary Search Tree &#8211; BST)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Cay-Nhi-Phan-Day-Du-Full-Binary-Tree\" >C\u00e2y Nh\u1ecb Ph\u00e2n \u0110\u1ea7y \u0110\u1ee7 (Full Binary Tree)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Cay-Nhi-Phan-Hoan-Chinh-Complete-Binary-Tree\" >C\u00e2y Nh\u1ecb Ph\u00e2n Ho\u00e0n Ch\u1ec9nh (Complete Binary Tree)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Ung-dung-co-ban-cua-Cay-Nhi-Phan-trong-thuc-te\" >\u1ee8ng d\u1ee5ng c\u01a1 b\u1ea3n c\u1ee7a C\u00e2y Nh\u1ecb Ph\u00e2n trong th\u1ef1c t\u1ebf<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/#Nhung-nham-lan-thuong-gap-khi-hoc-Cay-Nhi-Phan\" >Nh\u1eefng nh\u1ea7m l\u1eabn th\u01b0\u1eddng g\u1eb7p khi h\u1ecdc C\u00e2y Nh\u1ecb Ph\u00e2n<\/a><\/li><\/ul><\/nav><\/div>\n\n<p>N\u1ebfu b\u1ea1n \u0111ang h\u1ecdc <a href=\"https:\/\/interdata.vn\/blog\/cau-truc-du-lieu\/\">c\u1ea5u tr\u00fac d\u1eef li\u1ec7u<\/a> v\u00e0 <a href=\"https:\/\/interdata.vn\/blog\/thuat-toan-algorithm\/\">thu\u1eadt to\u00e1n<\/a>, th\u00ec kh\u00e1i ni\u1ec7m c\u00e2y nh\u1ecb ph\u00e2n (binary tree) ch\u1eafc ch\u1eafn l\u00e0 ch\u1ee7 \u0111\u1ec1 kh\u00f4ng th\u1ec3 b\u1ecf qua. V\u1edbi thi\u1ebft k\u1ebf \u0111\u01a1n gi\u1ea3n nh\u01b0ng c\u1ef1c k\u1ef3 hi\u1ec7u qu\u1ea3, c\u00e2y nh\u1ecb ph\u00e2n xu\u1ea5t hi\u1ec7n \u1edf m\u1ecdi n\u01a1i trong <a href=\"https:\/\/interdata.vn\/blog\/lap-trinh-la-gi\/\">l\u1eadp tr\u00ecnh<\/a>. Trong b\u00e0i vi\u1ebft n\u00e0y, b\u1ea1n s\u1ebd \u0111\u01b0\u1ee3c gi\u1ea3i th\u00edch t\u1eeb kh\u00e1i ni\u1ec7m c\u01a1 b\u1ea3n, c\u00e1c th\u00e0nh ph\u1ea7n c\u1ea5u th\u00e0nh c\u00e2y, ph\u00e2n lo\u1ea1i \u0111\u1ebfn \u1ee9ng d\u1ee5ng v\u00e0 v\u00ed d\u1ee5 minh h\u1ecda r\u00f5 r\u00e0ng, d\u1ec5 hi\u1ec3u.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Cay-nhi-phan-Binary-Tree-la-gi\"><\/span>C\u00e2y nh\u1ecb ph\u00e2n (Binary Tree) l\u00e0 g\u00ec?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong><a href=\"https:\/\/interdata.vn\/blog\/cay-nhi-phan-la-gi\/\">C\u00e2y nh\u1ecb ph\u00e2n (Binary Tree)<\/a><\/strong> l\u00e0 m\u1ed9t <strong>c\u1ea5u tr\u00fac d\u1eef li\u1ec7u<\/strong> ph\u00e2n c\u1ea5p, phi tuy\u1ebfn t\u00ednh, \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng r\u1ea5t ph\u1ed5 bi\u1ebfn trong <strong>khoa h\u1ecdc m\u00e1y t\u00ednh<\/strong>. \u0110i\u1ec3m \u0111\u1eb7c tr\u01b0ng nh\u1ea5t c\u1ee7a n\u00f3 l\u00e0 m\u1ed7i <strong>n\u00fat<\/strong> (node) trong c\u00e2y ch\u1ec9 c\u00f3 th\u1ec3 c\u00f3 t\u1ed1i \u0111a hai <strong>n\u00fat con<\/strong> (child node).<\/p>\n<p>Hai n\u00fat con n\u00e0y th\u01b0\u1eddng \u0111\u01b0\u1ee3c ph\u00e2n bi\u1ec7t r\u00f5 r\u00e0ng: m\u1ed9t <strong>n\u00fat con tr\u00e1i<\/strong> (left child) v\u00e0 m\u1ed9t <strong>n\u00fat con ph\u1ea3i<\/strong> (right child). Ch\u00ednh gi\u1edbi h\u1ea1n &#8220;t\u1ed1i \u0111a hai con&#8221; n\u00e0y \u0111\u00e3 t\u1ea1o n\u00ean t\u00ean g\u1ecdi &#8220;<strong>nh\u1ecb ph\u00e2n<\/strong>&#8221; (binary) cho c\u1ea5u tr\u00fac d\u1eef li\u1ec7u n\u00e0y.<\/p>\n<p>H\u00e3y t\u01b0\u1edfng t\u01b0\u1ee3ng c\u00e2y nh\u1ecb ph\u00e2n gi\u1ed1ng nh\u01b0 m\u1ed9t c\u00e2y gia ph\u1ea3 \u0111\u01a1n gi\u1ea3n, n\u01a1i m\u1ed7i ng\u01b0\u1eddi (n\u00fat) c\u00f3 t\u1ed1i \u0111a hai ng\u01b0\u1eddi con. C\u1ea5u tr\u00fac n\u00e0y b\u1eaft \u0111\u1ea7u t\u1eeb m\u1ed9t <strong>n\u00fat g\u1ed1c<\/strong> (root node) duy nh\u1ea5t \u1edf tr\u00ean c\u00f9ng v\u00e0 ph\u00e2n nh\u00e1nh xu\u1ed1ng d\u01b0\u1edbi, t\u1ea1o th\u00e0nh m\u1ed9t h\u1ec7 th\u1ed1ng th\u1ee9 b\u1eadc r\u00f5 r\u00e0ng.<\/p>\n<p>C\u00e1c th\u00e0nh ph\u1ea7n ch\u00ednh c\u1ea5u t\u1ea1o n\u00ean c\u00e2y nh\u1ecb ph\u00e2n l\u00e0 c\u00e1c <strong>n\u00fat<\/strong> (ch\u1ee9a th\u00f4ng tin) v\u00e0 c\u00e1c <strong>c\u1ea1nh<\/strong> (edge &#8211; \u0111\u01b0\u1eddng n\u1ed1i gi\u1eefa c\u00e1c n\u00fat). <strong>C\u1ea1nh<\/strong> th\u1ec3 hi\u1ec7n m\u1ed1i quan h\u1ec7 <strong>cha-con<\/strong> (parent-child), k\u1ebft n\u1ed1i m\u1ed9t n\u00fat cha v\u1edbi c\u00e1c n\u00fat con tr\u1ef1c ti\u1ebfp c\u1ee7a n\u00f3.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan.jpg\" alt=\"C\u00e2y nh\u1ecb ph\u00e2n\" width=\"750\" height=\"500\" class=\"aligncenter size-full wp-image-27390\" title=\"\" srcset=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan.jpg 750w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan-300x200.jpg 300w\" sizes=\"auto, (max-width: 750px) 100vw, 750px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Tai-sao-goi-la-%E2%80%9CNhi-phan%E2%80%9D\"><\/span>T\u1ea1i sao g\u1ecdi l\u00e0 &#8220;Nh\u1ecb ph\u00e2n&#8221;?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>T\u00ean g\u1ecdi &#8220;<strong>nh\u1ecb ph\u00e2n<\/strong>&#8221; b\u1eaft ngu\u1ed3n tr\u1ef1c ti\u1ebfp t\u1eeb \u0111\u1eb7c t\u00ednh c\u01a1 b\u1ea3n nh\u1ea5t c\u1ee7a c\u1ea5u tr\u00fac n\u00e0y. Trong ti\u1ebfng Latinh, &#8220;bi&#8221; c\u00f3 ngh\u0129a l\u00e0 hai. Do \u0111\u00f3, &#8220;nh\u1ecb ph\u00e2n&#8221; \u00e1m ch\u1ec9 r\u1eb1ng m\u1ed7i th\u00e0nh ph\u1ea7n (m\u1ed7i <strong>n\u00fat<\/strong>) trong c\u00e2y b\u1ecb gi\u1edbi h\u1ea1n \u1edf vi\u1ec7c c\u00f3 t\u1ed1i \u0111a hai nh\u00e1nh con tr\u1ef1c ti\u1ebfp \u0111i xu\u1ed1ng.<\/p>\n<p>\u0110i\u1ec1u n\u00e0y ph\u00e2n bi\u1ec7t <strong>c\u00e2y nh\u1ecb ph\u00e2n<\/strong> v\u1edbi c\u00e1c c\u1ea5u tr\u00fac c\u00e2y t\u1ed5ng qu\u00e1t h\u01a1n, n\u01a1i m\u1ed9t n\u00fat c\u00f3 th\u1ec3 c\u00f3 nhi\u1ec1u h\u01a1n hai n\u00fat con. V\u00ed d\u1ee5, khi b\u1ea1n xem c\u1ea5u tr\u00fac th\u01b0 m\u1ee5c tr\u00ean m\u00e1y t\u00ednh, m\u1ed9t th\u01b0 m\u1ee5c (n\u00fat cha) c\u00f3 th\u1ec3 ch\u1ee9a nhi\u1ec1u th\u01b0 m\u1ee5c con ho\u1eb7c t\u1ec7p tin (n\u00fat con).<\/p>\n<p>Gi\u1edbi h\u1ea1n hai con n\u00e0y mang l\u1ea1i s\u1ef1 \u0111\u01a1n gi\u1ea3n trong vi\u1ec7c qu\u1ea3n l\u00fd v\u00e0 <strong>bi\u1ec3u di\u1ec5n d\u1eef li\u1ec7u<\/strong>. N\u00f3 c\u0169ng l\u00e0 n\u1ec1n t\u1ea3ng cho nhi\u1ec1u thu\u1eadt to\u00e1n hi\u1ec7u qu\u1ea3, \u0111\u1eb7c bi\u1ec7t l\u00e0 trong vi\u1ec7c <strong>t\u00ecm ki\u1ebfm<\/strong> v\u00e0 <strong>s\u1eafp x\u1ebfp<\/strong> d\u1eef li\u1ec7u khi \u00e1p d\u1ee5ng c\u00e1c r\u00e0ng bu\u1ed9c c\u1ee5 th\u1ec3 (nh\u01b0 trong <strong>C\u00e2y Nh\u1ecb Ph\u00e2n T\u00ecm Ki\u1ebfm<\/strong>).<\/p>\n<p>V\u00ec v\u1eady, khi nghe \u0111\u1ebfn &#8220;c\u00e2y nh\u1ecb ph\u00e2n&#8221;, h\u00e3y nh\u1edb ngay \u0111\u1ebfn \u0111\u1eb7c \u0111i\u1ec3m c\u1ed1t l\u00f5i: m\u1ed7i n\u00fat c\u00f3 kh\u00f4ng qu\u00e1 hai con \u2013 m\u1ed9t b\u00ean tr\u00e1i v\u00e0 m\u1ed9t b\u00ean ph\u1ea3i. \u0110\u00e2y l\u00e0 ch\u00eca kh\u00f3a \u0111\u1ec3 hi\u1ec3u v\u00e0 l\u00e0m vi\u1ec7c v\u1edbi c\u1ea5u tr\u00fac d\u1eef li\u1ec7u quan tr\u1ecdng n\u00e0y.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Cac-thuat-ngu-quan-trong-can-biet-ve-Cay-Nhi-Phan\"><\/span>C\u00e1c thu\u1eadt ng\u1eef quan tr\u1ecdng c\u1ea7n bi\u1ebft v\u1ec1 C\u00e2y Nh\u1ecb Ph\u00e2n<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u0110\u1ec3 hi\u1ec3u s\u00e2u h\u01a1n v\u1ec1 <strong>c\u00e2y nh\u1ecb ph\u00e2n<\/strong>, ch\u00fang ta c\u1ea7n n\u1eafm v\u1eefng m\u1ed9t s\u1ed1 thu\u1eadt ng\u1eef c\u01a1 b\u1ea3n. Vi\u1ec7c n\u00e0y gi\u1ed1ng nh\u01b0 h\u1ecdc b\u1ea3ng ch\u1eef c\u00e1i tr\u01b0\u1edbc khi \u0111\u1ecdc m\u1ed9t cu\u1ed1n s\u00e1ch v\u1eady. \u0110\u1eebng lo, c\u00e1c kh\u00e1i ni\u1ec7m n\u00e0y kh\u00e1 tr\u1ef1c quan v\u00e0 d\u1ec5 hi\u1ec3u.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan-03.jpg\" alt=\"C\u00e2y nh\u1ecb ph\u00e2n 03\" width=\"750\" height=\"424\" class=\"aligncenter size-full wp-image-27389\" title=\"\" srcset=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan-03.jpg 750w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan-03-300x170.jpg 300w\" sizes=\"auto, (max-width: 750px) 100vw, 750px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Nut-Node\"><\/span>N\u00fat (Node)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>N\u00fat<\/strong> l\u00e0 th\u00e0nh ph\u1ea7n c\u01a1 b\u1ea3n nh\u1ea5t, \u0111\u01a1n v\u1ecb c\u1ea5u t\u1ea1o n\u00ean c\u00e2y nh\u1ecb ph\u00e2n. M\u1ed7i <strong>n\u00fat<\/strong> th\u01b0\u1eddng ch\u1ee9a m\u1ed9t gi\u00e1 tr\u1ecb d\u1eef li\u1ec7u n\u00e0o \u0111\u00f3 (v\u00ed d\u1ee5: m\u1ed9t con s\u1ed1, m\u1ed9t k\u00fd t\u1ef1, m\u1ed9t \u0111\u1ed1i t\u01b0\u1ee3ng) v\u00e0 c\u00e1c li\u00ean k\u1ebft (con tr\u1ecf ho\u1eb7c tham chi\u1ebfu) \u0111\u1ebfn c\u00e1c <strong>n\u00fat con<\/strong> c\u1ee7a n\u00f3 (n\u1ebfu c\u00f3).<\/p>\n<p>H\u00e3y h\u00ecnh dung m\u1ed7i <strong>n\u00fat<\/strong> nh\u01b0 m\u1ed9t chi\u1ebfc h\u1ed9p. B\u00ean trong h\u1ed9p ch\u1ee9a d\u1eef li\u1ec7u, v\u00e0 t\u1eeb h\u1ed9p \u0111\u00f3 c\u00f3 th\u1ec3 c\u00f3 t\u1ed1i \u0111a hai s\u1ee3i d\u00e2y n\u1ed1i \u0111\u1ebfn hai chi\u1ebfc h\u1ed9p kh\u00e1c \u1edf t\u1ea7ng d\u01b0\u1edbi \u2013 \u0111\u00f3 l\u00e0 c\u00e1c n\u00fat con tr\u00e1i v\u00e0 con ph\u1ea3i.<\/p>\n<p>Trong nhi\u1ec1u ng\u00f4n ng\u1eef <strong>l\u1eadp tr\u00ecnh<\/strong> nh\u01b0 C++, <a href=\"https:\/\/interdata.vn\/blog\/ngon-ngu-lap-trinh-java\/\">Java<\/a>, hay Python, m\u1ed9t <strong>n\u00fat<\/strong> th\u01b0\u1eddng \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n b\u1eb1ng m\u1ed9t l\u1edbp (class) ho\u1eb7c c\u1ea5u tr\u00fac (struct). N\u00f3 bao g\u1ed3m m\u1ed9t tr\u01b0\u1eddng \u0111\u1ec3 l\u01b0u d\u1eef li\u1ec7u v\u00e0 hai tr\u01b0\u1eddng con tr\u1ecf\/tham chi\u1ebfu <code>left<\/code> v\u00e0 <code>right<\/code> \u0111\u1ec3 ch\u1ec9 \u0111\u1ebfn c\u00e1c n\u00fat con t\u01b0\u01a1ng \u1ee9ng.<\/p>\n<p>Python<\/p>\n<pre><code class=\"language-plaintext\"># V\u00ed d\u1ee5 c\u1ea5u tr\u00fac N\u00fat \u0111\u01a1n gi\u1ea3n trong Python\r\nclass Node:\r\n    def __init__(self, data):\r\n        self.data = data  # Ch\u1ee9a d\u1eef li\u1ec7u c\u1ee7a n\u00fat\r\n        self.left = None  # Tham chi\u1ebfu \u0111\u1ebfn n\u00fat con tr\u00e1i, ban \u0111\u1ea7u l\u00e0 None\r\n        self.right = None # Tham chi\u1ebfu \u0111\u1ebfn n\u00fat con ph\u1ea3i, ban \u0111\u1ea7u l\u00e0 None\r\n<\/code><\/pre>\n<p>\u0110o\u1ea1n m\u00e3 tr\u00ean minh h\u1ecda m\u1ed9t c\u00e1ch \u0111\u01a1n gi\u1ea3n c\u1ea5u tr\u00fac c\u1ee7a m\u1ed9t <strong>n\u00fat<\/strong>. M\u1ed7i khi t\u1ea1o m\u1ed9t n\u00fat m\u1edbi, b\u1ea1n cung c\u1ea5p d\u1eef li\u1ec7u cho n\u00f3, v\u00e0 ban \u0111\u1ea7u, n\u00f3 ch\u01b0a c\u00f3 con n\u00e0o (<code>left<\/code> v\u00e0 <code>right<\/code> \u0111\u1ec1u l\u00e0 <code>None<\/code>).<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Canh-Edge\"><\/span>C\u1ea1nh (Edge)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>C\u1ea1nh<\/strong> l\u00e0 \u0111\u01b0\u1eddng n\u1ed1i tr\u1ef1c ti\u1ebfp gi\u1eefa hai <strong>n\u00fat<\/strong> trong c\u00e2y, th\u1ec3 hi\u1ec7n m\u1ed1i quan h\u1ec7 <strong>cha-con<\/strong>. C\u1ee5 th\u1ec3, m\u1ed9t <strong>c\u1ea1nh<\/strong> s\u1ebd n\u1ed1i t\u1eeb m\u1ed9t <strong>n\u00fat cha<\/strong> (parent node) \u0111\u1ebfn m\u1ed9t <strong>n\u00fat con<\/strong> (child node) c\u1ee7a n\u00f3.<\/p>\n<p>Trong c\u00e1c m\u00f4 h\u00ecnh c\u00e2y, <strong>c\u1ea1nh<\/strong> th\u01b0\u1eddng \u0111\u01b0\u1ee3c v\u1ebd d\u01b0\u1edbi d\u1ea1ng m\u1ed9t \u0111o\u1ea1n th\u1eb3ng n\u1ed1i hai n\u00fat. N\u00f3 bi\u1ec3u th\u1ecb m\u1ed9t li\u00ean k\u1ebft ho\u1eb7c m\u1ed9t con tr\u1ecf t\u1eeb n\u00fat cha \u0111\u1ebfn n\u00fat con. S\u1ed1 l\u01b0\u1ee3ng c\u1ea1nh trong m\u1ed9t c\u00e2y lu\u00f4n \u00edt h\u01a1n s\u1ed1 l\u01b0\u1ee3ng n\u00fat \u0111\u00fang m\u1ed9t \u0111\u01a1n v\u1ecb (S\u1ed1 c\u1ea1nh = S\u1ed1 n\u00fat &#8211; 1).<\/p>\n<p>V\u00ed d\u1ee5, n\u1ebfu n\u00fat A l\u00e0 cha c\u1ee7a n\u00fat B, th\u00ec s\u1ebd c\u00f3 m\u1ed9t <strong>c\u1ea1nh<\/strong> n\u1ed1i t\u1eeb A xu\u1ed1ng B. C\u1ea1nh n\u00e0y cho th\u1ea5y B l\u00e0 h\u1eadu du\u1ec7 tr\u1ef1c ti\u1ebfp c\u1ee7a A. C\u00e1c c\u1ea1nh gi\u00fap x\u00e1c \u0111\u1ecbnh c\u1ea5u tr\u00fac ph\u00e2n c\u1ea5p v\u00e0 \u0111\u01b0\u1eddng \u0111i trong c\u00e2y.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Nut-Goc-Root-Node\"><\/span>N\u00fat G\u1ed1c (Root Node)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>N\u00fat g\u1ed1c<\/strong> l\u00e0 <strong>n\u00fat<\/strong> \u0111\u1eb7c bi\u1ec7t nh\u1ea5t trong c\u00e2y nh\u1ecb ph\u00e2n. \u0110\u00e2y l\u00e0 n\u00fat n\u1eb1m \u1edf v\u1ecb tr\u00ed cao nh\u1ea5t, l\u00e0 \u0111i\u1ec3m kh\u1edfi \u0111\u1ea7u c\u1ee7a c\u00e2y v\u00e0 <strong>kh\u00f4ng c\u00f3 n\u00fat cha n\u00e0o<\/strong>. M\u1ed7i c\u00e2y nh\u1ecb ph\u00e2n (n\u1ebfu kh\u00f4ng r\u1ed7ng) ch\u1ec9 c\u00f3 duy nh\u1ea5t m\u1ed9t <strong>n\u00fat g\u1ed1c<\/strong>.<\/p>\n<p>H\u00e3y t\u01b0\u1edfng t\u01b0\u1ee3ng <strong>n\u00fat g\u1ed1c<\/strong> nh\u01b0 g\u1ed1c c\u1ee7a m\u1ed9t c\u00e1i c\u00e2y th\u1eadt trong t\u1ef1 nhi\u00ean, t\u1eeb \u0111\u00f3 m\u1ecdi c\u00e0nh nh\u00e1nh kh\u00e1c ph\u00e1t tri\u1ec3n ra. T\u1ea5t c\u1ea3 c\u00e1c n\u00fat kh\u00e1c trong c\u00e2y \u0111\u1ec1u l\u00e0 h\u1eadu du\u1ec7 (con, ch\u00e1u, ch\u1eaft&#8230;) c\u1ee7a <strong>n\u00fat g\u1ed1c<\/strong>.<\/p>\n<p>N\u1ebfu m\u1ed9t c\u00e2y nh\u1ecb ph\u00e2n ch\u1ec9 c\u00f3 \u0111\u00fang m\u1ed9t n\u00fat, th\u00ec n\u00fat \u0111\u00f3 v\u1eeba l\u00e0 <strong>n\u00fat g\u1ed1c<\/strong>, v\u1eeba l\u00e0 <strong>n\u00fat l\u00e1<\/strong>. Vi\u1ec7c x\u00e1c \u0111\u1ecbnh <strong>n\u00fat g\u1ed1c<\/strong> l\u00e0 b\u01b0\u1edbc \u0111\u1ea7u ti\u00ean trong h\u1ea7u h\u1ebft c\u00e1c thao t\u00e1c x\u1eed l\u00fd tr\u00ean c\u00e2y, ch\u1eb3ng h\u1ea1n nh\u01b0 <strong>duy\u1ec7t c\u00e2y<\/strong> (tree traversal).<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Nut-La-Leaf-Node\"><\/span>N\u00fat L\u00e1 (Leaf Node)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>N\u00fat l\u00e1<\/strong> l\u00e0 nh\u1eefng <strong>n\u00fat<\/strong> n\u1eb1m \u1edf v\u1ecb tr\u00ed &#8220;t\u1eadn c\u00f9ng&#8221; c\u1ee7a c\u00e1c nh\u00e1nh c\u00e2y. \u0110\u1eb7c \u0111i\u1ec3m nh\u1eadn d\u1ea1ng c\u1ee7a <strong>n\u00fat l\u00e1<\/strong> l\u00e0 ch\u00fang <strong>kh\u00f4ng c\u00f3 b\u1ea5t k\u1ef3 n\u00fat con n\u00e0o<\/strong> (c\u1ea3 con tr\u00e1i v\u00e0 con ph\u1ea3i \u0111\u1ec1u l\u00e0 <code>None<\/code> ho\u1eb7c <code>NULL<\/code>).<\/p>\n<p>Trong s\u01a1 \u0111\u1ed3 c\u00e2y, c\u00e1c <strong>n\u00fat l\u00e1<\/strong> th\u01b0\u1eddng n\u1eb1m \u1edf t\u1ea7ng th\u1ea5p nh\u1ea5t. Ch\u00fang \u0111\u1ea1i di\u1ec7n cho \u0111i\u1ec3m k\u1ebft th\u00fac c\u1ee7a c\u00e1c \u0111\u01b0\u1eddng \u0111i t\u1eeb <strong>n\u00fat g\u1ed1c<\/strong>. M\u1ed9t c\u00e2y nh\u1ecb ph\u00e2n c\u00f3 th\u1ec3 c\u00f3 m\u1ed9t ho\u1eb7c nhi\u1ec1u <strong>n\u00fat l\u00e1<\/strong>.<\/p>\n<p>V\u00ed d\u1ee5, trong c\u00e2y gia ph\u1ea3, nh\u1eefng ng\u01b0\u1eddi kh\u00f4ng c\u00f3 con c\u00e1i s\u1ebd t\u01b0\u01a1ng \u1ee9ng v\u1edbi c\u00e1c <strong>n\u00fat l\u00e1<\/strong>. Vi\u1ec7c x\u00e1c \u0111\u1ecbnh c\u00e1c <strong>n\u00fat l\u00e1<\/strong> r\u1ea5t quan tr\u1ecdng trong nhi\u1ec1u thu\u1eadt to\u00e1n, v\u00ed d\u1ee5 nh\u01b0 t\u00ednh to\u00e1n <strong>chi\u1ec1u cao c\u1ee7a c\u00e2y<\/strong>.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Nut-Cha-Parent-Node-Nut-Con-Child-Node\"><\/span>N\u00fat Cha (Parent Node) &amp; N\u00fat Con (Child Node)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>M\u1ed1i quan h\u1ec7 <strong>cha-con<\/strong> l\u00e0 c\u1ed1t l\u00f5i c\u1ee7a c\u1ea5u tr\u00fac ph\u00e2n c\u1ea5p trong c\u00e2y nh\u1ecb ph\u00e2n. M\u1ed9t <strong>n\u00fat cha<\/strong> l\u00e0 n\u00fat c\u00f3 c\u00e1c <strong>c\u1ea1nh<\/strong> n\u1ed1i tr\u1ef1c ti\u1ebfp xu\u1ed1ng c\u00e1c n\u00fat kh\u00e1c. Nh\u1eefng n\u00fat \u0111\u01b0\u1ee3c n\u1ed1i t\u1edbi \u0111\u00f3 \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 <strong>n\u00fat con<\/strong> c\u1ee7a n\u00f3.<\/p>\n<p>Nh\u01b0 \u0111\u00e3 \u0111\u1ec1 c\u1eadp, trong c\u00e2y nh\u1ecb ph\u00e2n, m\u1ed7i <strong>n\u00fat cha<\/strong> c\u00f3 th\u1ec3 c\u00f3 t\u1ed1i \u0111a hai <strong>n\u00fat con<\/strong>: m\u1ed9t <strong>n\u00fat con tr\u00e1i<\/strong> v\u00e0 m\u1ed9t <strong>n\u00fat con ph\u1ea3i<\/strong>. Ng\u01b0\u1ee3c l\u1ea1i, m\u1ed7i n\u00fat (tr\u1eeb <strong>n\u00fat g\u1ed1c<\/strong>) ch\u1ec9 c\u00f3 duy nh\u1ea5t m\u1ed9t <strong>n\u00fat cha<\/strong>.<\/p>\n<p>M\u1ed1i quan h\u1ec7 n\u00e0y t\u1ea1o ra m\u1ed9t c\u1ea5u tr\u00fac th\u1ee9 b\u1eadc r\u00f5 r\u00e0ng. N\u00fat A l\u00e0 cha c\u1ee7a n\u00fat B n\u1ebfu c\u00f3 m\u1ed9t c\u1ea1nh n\u1ed1i tr\u1ef1c ti\u1ebfp t\u1eeb A \u0111\u1ebfn B. Ng\u01b0\u1ee3c l\u1ea1i, B l\u00e0 con c\u1ee7a A. Hi\u1ec3u r\u00f5 m\u1ed1i quan h\u1ec7 n\u00e0y gi\u00fap ch\u00fang ta di chuy\u1ec3n v\u00e0 thao t\u00e1c tr\u00ean c\u00e2y.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Nut-Trung-Gian-Internal-Node\"><\/span>N\u00fat Trung Gian (Internal Node)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>N\u00fat trung gian<\/strong> l\u00e0 b\u1ea5t k\u1ef3 <strong>n\u00fat<\/strong> n\u00e0o trong c\u00e2y m\u00e0 <strong>kh\u00f4ng ph\u1ea3i l\u00e0 n\u00fat l\u00e1<\/strong>. \u0110i\u1ec1u n\u00e0y c\u00f3 ngh\u0129a l\u00e0 m\u1ed9t <strong>n\u00fat trung gian<\/strong> ph\u1ea3i c\u00f3 \u00edt nh\u1ea5t m\u1ed9t <strong>n\u00fat con<\/strong> (con tr\u00e1i, con ph\u1ea3i, ho\u1eb7c c\u1ea3 hai).<\/p>\n<p><strong>N\u00fat g\u1ed1c<\/strong> c\u0169ng \u0111\u01b0\u1ee3c xem l\u00e0 m\u1ed9t <strong>n\u00fat trung gian<\/strong> n\u1ebfu n\u00f3 c\u00f3 \u00edt nh\u1ea5t m\u1ed9t n\u00fat con (t\u1ee9c l\u00e0 c\u00e2y c\u00f3 nhi\u1ec1u h\u01a1n m\u1ed9t n\u00fat). C\u00e1c <strong>n\u00fat trung gian<\/strong> \u0111\u00f3ng vai tr\u00f2 k\u1ebft n\u1ed1i, ph\u00e2n nh\u00e1nh trong c\u1ea5u tr\u00fac c\u00e2y.<\/p>\n<p>T\u00f3m l\u1ea1i, c\u00e1c n\u00fat trong c\u00e2y nh\u1ecb ph\u00e2n c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c ph\u00e2n lo\u1ea1i th\u00e0nh: <strong>n\u00fat g\u1ed1c<\/strong>, <strong>n\u00fat trung gian<\/strong> (kh\u00f4ng ph\u1ea3i g\u1ed1c v\u00e0 kh\u00f4ng ph\u1ea3i l\u00e1), v\u00e0 <strong>n\u00fat l\u00e1<\/strong>. T\u1ed5ng s\u1ed1 n\u00fat trong c\u00e2y b\u1eb1ng t\u1ed5ng s\u1ed1 n\u00fat c\u1ee7a ba lo\u1ea1i n\u00e0y.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Cay-con-Subtree\"><\/span>C\u00e2y con (Subtree)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>M\u1ed9t <strong>c\u00e2y con<\/strong> c\u1ee7a m\u1ed9t c\u00e2y nh\u1ecb ph\u00e2n l\u00e0 m\u1ed9t <strong>n\u00fat<\/strong> trong c\u00e2y \u0111\u00f3 c\u00f9ng v\u1edbi t\u1ea5t c\u1ea3 c\u00e1c h\u1eadu du\u1ec7 c\u1ee7a n\u00f3 (con, ch\u00e1u, ch\u1eaft&#8230;). B\u1ea3n th\u00e2n <strong>c\u00e2y con<\/strong> c\u0169ng l\u00e0 m\u1ed9t c\u00e2y nh\u1ecb ph\u00e2n ho\u00e0n ch\u1ec9nh.<\/p>\n<p>M\u1ed7i n\u00fat trong c\u00e2y nh\u1ecb ph\u00e2n (tr\u1eeb c\u00e1c n\u00fat l\u00e1) l\u00e0 g\u1ed1c c\u1ee7a hai <strong>c\u00e2y con<\/strong> ti\u1ec1m n\u0103ng: <strong>c\u00e2y con tr\u00e1i<\/strong> v\u00e0 <strong>c\u00e2y con ph\u1ea3i<\/strong>. <strong>C\u00e2y con tr\u00e1i<\/strong> bao g\u1ed3m n\u00fat con tr\u00e1i v\u00e0 t\u1ea5t c\u1ea3 h\u1eadu du\u1ec7 c\u1ee7a n\u00f3. T\u01b0\u01a1ng t\u1ef1 v\u1edbi <strong>c\u00e2y con ph\u1ea3i<\/strong>.<\/p>\n<p>Kh\u00e1i ni\u1ec7m <strong>c\u00e2y con<\/strong> r\u1ea5t quan tr\u1ecdng v\u00ec n\u00f3 th\u1ec3 hi\u1ec7n t\u00ednh ch\u1ea5t \u0111\u1ec7 quy c\u1ee7a c\u00e2y nh\u1ecb ph\u00e2n. Nhi\u1ec1u thu\u1eadt to\u00e1n x\u1eed l\u00fd c\u00e2y ho\u1ea1t \u0111\u1ed9ng d\u1ef1a tr\u00ean nguy\u00ean t\u1eafc gi\u1ea3i quy\u1ebft b\u00e0i to\u00e1n tr\u00ean c\u00e1c <strong>c\u00e2y con<\/strong> r\u1ed3i k\u1ebft h\u1ee3p k\u1ebft qu\u1ea3 l\u1ea1i.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Do-sau-cua-Nut-Depth-of-a-Node\"><\/span>\u0110\u1ed9 s\u00e2u c\u1ee7a N\u00fat (Depth of a Node)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>\u0110\u1ed9 s\u00e2u<\/strong> c\u1ee7a m\u1ed9t <strong>n\u00fat<\/strong> \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a l\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng \u0111i (s\u1ed1 l\u01b0\u1ee3ng <strong>c\u1ea1nh<\/strong>) t\u1eeb <strong>n\u00fat g\u1ed1c<\/strong> \u0111\u1ebfn n\u00fat \u0111\u00f3. Theo quy \u01b0\u1edbc, <strong>\u0111\u1ed9 s\u00e2u c\u1ee7a n\u00fat g\u1ed1c<\/strong> l\u00e0 0.<\/p>\n<p>C\u00e1c n\u00fat con tr\u1ef1c ti\u1ebfp c\u1ee7a <strong>n\u00fat g\u1ed1c<\/strong> s\u1ebd c\u00f3 <strong>\u0111\u1ed9 s\u00e2u<\/strong> l\u00e0 1. C\u00e1c n\u00fat con c\u1ee7a nh\u1eefng n\u00fat c\u00f3 \u0111\u1ed9 s\u00e2u 1 s\u1ebd c\u00f3 <strong>\u0111\u1ed9 s\u00e2u<\/strong> l\u00e0 2, v\u00e0 c\u1ee9 th\u1ebf ti\u1ebfp t\u1ee5c. <strong>\u0110\u1ed9 s\u00e2u<\/strong> cho bi\u1ebft &#8220;m\u1ee9c&#8221; (level) c\u1ee7a m\u1ed9t n\u00fat trong c\u00e2y.<\/p>\n<p>V\u00ed d\u1ee5, n\u1ebfu \u0111\u01b0\u1eddng \u0111i t\u1eeb g\u1ed1c A \u0111\u1ebfn n\u00fat D l\u00e0 A -&gt; B -&gt; D, th\u00ec \u0111\u01b0\u1eddng \u0111i n\u00e0y c\u00f3 2 c\u1ea1nh (A-&gt;B v\u00e0 B-&gt;D). Do \u0111\u00f3, <strong>\u0111\u1ed9 s\u00e2u<\/strong> c\u1ee7a n\u00fat D l\u00e0 2. Kh\u00e1i ni\u1ec7m n\u00e0y h\u1eefu \u00edch khi ph\u00e2n t\u00edch c\u1ea5u tr\u00fac theo t\u1eebng t\u1ea7ng c\u1ee7a c\u00e2y.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Chieu-cao-cua-Cay-Height-of-a-Tree\"><\/span>Chi\u1ec1u cao c\u1ee7a C\u00e2y (Height of a Tree)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Chi\u1ec1u cao<\/strong> c\u1ee7a m\u1ed9t c\u00e2y nh\u1ecb ph\u00e2n \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a l\u00e0 <strong>\u0111\u1ed9 s\u00e2u<\/strong> l\u1edbn nh\u1ea5t c\u1ee7a m\u1ed9t <strong>n\u00fat l\u00e1<\/strong> trong c\u00e2y \u0111\u00f3. N\u00f3i c\u00e1ch kh\u00e1c, \u0111\u00f3 l\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng \u0111i d\u00e0i nh\u1ea5t (t\u00ednh b\u1eb1ng s\u1ed1 c\u1ea1nh) t\u1eeb <strong>n\u00fat g\u1ed1c<\/strong> \u0111\u1ebfn m\u1ed9t <strong>n\u00fat l\u00e1<\/strong> b\u1ea5t k\u1ef3.<\/p>\n<p>Theo m\u1ed9t quy \u01b0\u1edbc kh\u00e1c (\u0111\u00f4i khi d\u00f9ng s\u1ed1 n\u00fat thay v\u00ec s\u1ed1 c\u1ea1nh), <strong>chi\u1ec1u cao<\/strong> c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c t\u00ednh b\u1eb1ng s\u1ed1 n\u00fat tr\u00ean \u0111\u01b0\u1eddng \u0111i d\u00e0i nh\u1ea5t t\u1eeb g\u1ed1c \u0111\u1ebfn l\u00e1. N\u1ebfu t\u00ednh theo s\u1ed1 c\u1ea1nh, chi\u1ec1u cao c\u1ee7a c\u00e2y ch\u1ec9 c\u00f3 m\u1ed9t n\u00fat (g\u1ed1c) l\u00e0 0. N\u1ebfu t\u00ednh theo s\u1ed1 n\u00fat, chi\u1ec1u cao l\u00e0 1. Ch\u00fang ta th\u01b0\u1eddng d\u00f9ng \u0111\u1ecbnh ngh\u0129a theo s\u1ed1 c\u1ea1nh.<\/p>\n<p><strong>Chi\u1ec1u cao<\/strong> cho bi\u1ebft &#8220;\u0111\u1ed9 v\u01b0\u01a1n xa&#8221; c\u1ee7a c\u00e2y t\u1eeb g\u1ed1c. V\u00ed d\u1ee5, n\u1ebfu \u0111\u01b0\u1eddng \u0111i d\u00e0i nh\u1ea5t t\u1eeb g\u1ed1c \u0111\u1ebfn l\u00e1 c\u00f3 3 c\u1ea1nh, th\u00ec <strong>chi\u1ec1u cao<\/strong> c\u1ee7a c\u00e2y l\u00e0 3. <strong>Chi\u1ec1u cao<\/strong> l\u00e0 m\u1ed9t th\u00f4ng s\u1ed1 quan tr\u1ecdng \u0111\u1ec3 \u0111\u00e1nh gi\u00e1 hi\u1ec7u su\u1ea5t c\u1ee7a c\u00e1c thu\u1eadt to\u00e1n tr\u00ean c\u00e2y, \u0111\u1eb7c bi\u1ec7t l\u00e0 <strong>c\u00e2y nh\u1ecb ph\u00e2n t\u00ecm ki\u1ebfm<\/strong>.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Bac-cua-Nut-Degree-of-a-Node\"><\/span>B\u1eadc c\u1ee7a N\u00fat (Degree of a Node)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>B\u1eadc<\/strong> c\u1ee7a m\u1ed9t <strong>n\u00fat<\/strong> \u0111\u01a1n gi\u1ea3n l\u00e0 s\u1ed1 l\u01b0\u1ee3ng <strong>n\u00fat con<\/strong> m\u00e0 n\u00fat \u0111\u00f3 c\u00f3. Trong c\u00e2y nh\u1ecb ph\u00e2n, <strong>b\u1eadc<\/strong> c\u1ee7a m\u1ed9t n\u00fat ch\u1ec9 c\u00f3 th\u1ec3 l\u00e0 0, 1 ho\u1eb7c 2.<\/p>\n<ul>\n<li><strong>B\u1eadc 0:<\/strong> N\u00fat kh\u00f4ng c\u00f3 con n\u00e0o (\u0111\u00e2y l\u00e0 <strong>n\u00fat l\u00e1<\/strong>).<\/li>\n<li><strong>B\u1eadc 1:<\/strong> N\u00fat ch\u1ec9 c\u00f3 m\u1ed9t con (ho\u1eb7c con tr\u00e1i, ho\u1eb7c con ph\u1ea3i).<\/li>\n<li><strong>B\u1eadc 2:<\/strong> N\u00fat c\u00f3 \u0111\u1ee7 c\u1ea3 hai con (con tr\u00e1i v\u00e0 con ph\u1ea3i).<\/li>\n<\/ul>\n<p><strong>B\u1eadc<\/strong> c\u1ee7a m\u1ed9t n\u00fat cho bi\u1ebft kh\u1ea3 n\u0103ng ph\u00e2n nh\u00e1nh t\u1ea1i n\u00fat \u0111\u00f3. Kh\u00e1i ni\u1ec7m n\u00e0y gi\u00fap ph\u00e2n lo\u1ea1i c\u00e1c n\u00fat trong c\u00e2y (v\u00ed d\u1ee5: n\u00fat l\u00e1 c\u00f3 b\u1eadc 0, n\u00fat \u0111\u1ea7y \u0111\u1ee7 c\u00f3 b\u1eadc 2).<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Dac-diem-va-tinh-chat-co-ban-cua-Cay-Nhi-Phan\"><\/span>\u0110\u1eb7c \u0111i\u1ec3m v\u00e0 t\u00ednh ch\u1ea5t c\u01a1 b\u1ea3n c\u1ee7a C\u00e2y Nh\u1ecb Ph\u00e2n<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Sau khi n\u1eafm v\u1eefng c\u00e1c thu\u1eadt ng\u1eef, ch\u00fang ta c\u00f9ng t\u1ed5ng k\u1ebft l\u1ea1i m\u1ed9t s\u1ed1 <strong>\u0111\u1eb7c \u0111i\u1ec3m<\/strong> v\u00e0 <strong>t\u00ednh ch\u1ea5t<\/strong> quan tr\u1ecdng c\u1ee7a <strong>c\u00e2y nh\u1ecb ph\u00e2n<\/strong>:<\/p>\n<ol>\n<li><strong>C\u1ea5u tr\u00fac ph\u00e2n c\u1ea5p (Hierarchical Structure):<\/strong> D\u1eef li\u1ec7u \u0111\u01b0\u1ee3c t\u1ed5 ch\u1ee9c theo c\u00e1c m\u1ee9c, b\u1eaft \u0111\u1ea7u t\u1eeb <strong>n\u00fat g\u1ed1c<\/strong>, th\u1ec3 hi\u1ec7n m\u1ed1i quan h\u1ec7 cha-con r\u00f5 r\u00e0ng. \u0110\u00e2y l\u00e0 c\u1ea5u tr\u00fac <strong>phi tuy\u1ebfn t\u00ednh<\/strong>, kh\u00e1c v\u1edbi m\u1ea3ng hay danh s\u00e1ch li\u00ean k\u1ebft.<\/li>\n<li><strong>Gi\u1edbi h\u1ea1n hai con:<\/strong> M\u1ed7i <strong>n\u00fat<\/strong> c\u00f3 t\u1ed1i \u0111a hai <strong>n\u00fat con<\/strong>, \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 con tr\u00e1i v\u00e0 con ph\u1ea3i. \u0110\u00e2y l\u00e0 <strong>\u0111\u1eb7c \u0111i\u1ec3m<\/strong> \u0111\u1ecbnh danh c\u1ee7a <strong>c\u00e2y nh\u1ecb ph\u00e2n<\/strong>.<\/li>\n<li><strong>Th\u1ee9 t\u1ef1 con (Optional):<\/strong> Trong c\u00e2y nh\u1ecb ph\u00e2n n\u00f3i chung, th\u1ee9 t\u1ef1 c\u1ee7a con tr\u00e1i v\u00e0 con ph\u1ea3i l\u00e0 quan tr\u1ecdng. C\u00e2y c\u00f3 n\u00fat A l\u00e0 g\u1ed1c, B l\u00e0 con tr\u00e1i kh\u00e1c v\u1edbi c\u00e2y c\u00f3 A l\u00e0 g\u1ed1c, B l\u00e0 con ph\u1ea3i. Tuy nhi\u00ean, trong m\u1ed9t s\u1ed1 \u1ee9ng d\u1ee5ng c\u1ee5 th\u1ec3 nh\u01b0 <strong>C\u00e2y Nh\u1ecb Ph\u00e2n T\u00ecm Ki\u1ebfm<\/strong>, th\u1ee9 t\u1ef1 n\u00e0y c\u00f2n mang th\u00eam \u00fd ngh\u0129a v\u1ec1 gi\u00e1 tr\u1ecb d\u1eef li\u1ec7u.<\/li>\n<li><strong>T\u00ednh \u0111\u1ec7 quy (Recursive Nature):<\/strong> M\u1ed7i <strong>c\u00e2y con<\/strong> (tr\u00e1i ho\u1eb7c ph\u1ea3i) c\u1ee7a m\u1ed9t n\u00fat trong c\u00e2y nh\u1ecb ph\u00e2n c\u0169ng ch\u00ednh l\u00e0 m\u1ed9t c\u00e2y nh\u1ecb ph\u00e2n. T\u00ednh ch\u1ea5t n\u00e0y cho ph\u00e9p \u00e1p d\u1ee5ng c\u00e1c gi\u1ea3i thu\u1eadt \u0111\u1ec7 quy m\u1ed9t c\u00e1ch t\u1ef1 nhi\u00ean v\u00e0 hi\u1ec7u qu\u1ea3.<\/li>\n<li><strong>M\u1ed1i quan h\u1ec7 gi\u1eefa S\u1ed1 n\u00fat v\u00e0 Chi\u1ec1u cao:<\/strong> <strong>Chi\u1ec1u cao<\/strong> (h) c\u1ee7a c\u00e2y nh\u1ecb ph\u00e2n c\u00f3 N n\u00fat n\u1eb1m trong kho\u1ea3ng: <code>floor(log2(N)) &lt;= h &lt;= N-1<\/code>. Chi\u1ec1u cao t\u1ed1i thi\u1ec3u \u0111\u1ea1t \u0111\u01b0\u1ee3c \u1edf c\u00e2y g\u1ea7n ho\u00e0n ch\u1ec9nh, v\u00e0 chi\u1ec1u cao t\u1ed1i \u0111a x\u1ea3y ra \u1edf c\u00e2y suy bi\u1ebfn (gi\u1ed1ng danh s\u00e1ch li\u00ean k\u1ebft).<\/li>\n<li><strong>S\u1ed1 n\u00fat t\u1ed1i \u0111a \u1edf m\u1ed9t m\u1ee9c:<\/strong> S\u1ed1 l\u01b0\u1ee3ng n\u00fat t\u1ed1i \u0111a \u1edf m\u1ee9c <code>k<\/code> (v\u1edbi g\u1ed1c \u1edf m\u1ee9c 0) l\u00e0 2k. \u0110i\u1ec1u n\u00e0y c\u00f3 ngh\u0129a l\u00e0 s\u1ed1 l\u01b0\u1ee3ng n\u00fat t\u0103ng g\u1ea5p \u0111\u00f4i \u1edf m\u1ed7i m\u1ee9c ti\u1ebfp theo trong tr\u01b0\u1eddng h\u1ee3p c\u00e2y &#8220;\u0111\u1ea7y&#8221; nh\u1ea5t c\u00f3 th\u1ec3.<\/li>\n<li><strong>T\u1ed5ng s\u1ed1 n\u00fat t\u1ed1i \u0111a:<\/strong> M\u1ed9t c\u00e2y nh\u1ecb ph\u00e2n c\u00f3 <strong>chi\u1ec1u cao<\/strong> <code>h<\/code> (t\u00ednh theo s\u1ed1 c\u1ea1nh, g\u1ed1c c\u00f3 chi\u1ec1u cao 0) c\u00f3 th\u1ec3 ch\u1ee9a t\u1ed1i \u0111a 2h+1\u22121 n\u00fat. \u0110i\u1ec1u n\u00e0y x\u1ea3y ra khi c\u00e2y l\u00e0 m\u1ed9t <strong>c\u00e2y nh\u1ecb ph\u00e2n \u0111\u1ea7y \u0111\u1ee7<\/strong> v\u00e0 t\u1ea5t c\u1ea3 c\u00e1c m\u1ee9c \u0111\u1ec1u \u0111\u01b0\u1ee3c l\u1ea5p \u0111\u1ea7y.<\/li>\n<\/ol>\n<p>Nh\u1eefng <strong>\u0111\u1eb7c \u0111i\u1ec3m<\/strong> v\u00e0 <strong>t\u00ednh ch\u1ea5t<\/strong> n\u00e0y l\u00e0m cho <strong>c\u00e2y nh\u1ecb ph\u00e2n<\/strong> tr\u1edf th\u00e0nh m\u1ed9t c\u00f4ng c\u1ee5 m\u1ea1nh m\u1ebd v\u00e0 linh ho\u1ea1t \u0111\u1ec3 <strong>bi\u1ec3u di\u1ec5n d\u1eef li\u1ec7u<\/strong> v\u00e0 thi\u1ebft k\u1ebf c\u00e1c <strong>thu\u1eadt to\u00e1n<\/strong> hi\u1ec7u qu\u1ea3 trong nhi\u1ec1u l\u0129nh v\u1ef1c c\u1ee7a <strong>khoa h\u1ecdc m\u00e1y t\u00ednh<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan-01.jpg\" alt=\"C\u00e2y nh\u1ecb ph\u00e2n 01\" width=\"750\" height=\"465\" class=\"aligncenter size-full wp-image-27387\" title=\"\" srcset=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan-01.jpg 750w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan-01-300x186.jpg 300w\" sizes=\"auto, (max-width: 750px) 100vw, 750px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Vi-du-minh-hoa-ve-Cay-Nhi-Phan-don-gian\"><\/span>V\u00ed d\u1ee5 minh h\u1ecda v\u1ec1 C\u00e2y Nh\u1ecb Ph\u00e2n \u0111\u01a1n gi\u1ea3n<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>L\u00fd thuy\u1ebft s\u1ebd d\u1ec5 hi\u1ec3u h\u01a1n r\u1ea5t nhi\u1ec1u khi c\u00f3 <strong>v\u00ed d\u1ee5<\/strong> th\u1ef1c t\u1ebf. H\u00e3y c\u00f9ng xem x\u00e9t m\u1ed9t <strong>c\u00e2y nh\u1ecb ph\u00e2n<\/strong> \u0111\u01a1n gi\u1ea3n sau \u0111\u00e2y. Gi\u1ea3 s\u1eed ch\u00fang ta c\u00f3 c\u00e1c n\u00fat ch\u1ee9a c\u00e1c gi\u00e1 tr\u1ecb <a href=\"https:\/\/interdata.vn\/blog\/so-nguyen-integer\/\">s\u1ed1 nguy\u00ean<\/a>:<\/p>\n<pre><code class=\"language-plaintext\">      10 (G\u1ed1c)\r\n     \/  \\\r\n    5    15\r\n   \/ \\    \\\r\n  3   7    20 (L\u00e1)\r\n \/       \/\r\n1 (L\u00e1)  18 (L\u00e1)\r\n<\/code><\/pre>\n<p>H\u00e3y c\u00f9ng ph\u00e2n t\u00edch c\u00e2y n\u00e0y d\u1ef1a tr\u00ean c\u00e1c thu\u1eadt ng\u1eef \u0111\u00e3 h\u1ecdc:<\/p>\n<ul>\n<li><strong>N\u00fat G\u1ed1c:<\/strong> N\u00fat c\u00f3 gi\u00e1 tr\u1ecb 10.<\/li>\n<li><strong>C\u00e1c N\u00fat:<\/strong> 10, 5, 15, 3, 7, 20, 1, 18. T\u1ed5ng c\u1ed9ng c\u00f3 8 n\u00fat.<\/li>\n<li><strong>C\u00e1c C\u1ea1nh:<\/strong> C\u00f3 7 c\u1ea1nh n\u1ed1i c\u00e1c n\u00fat (10-5, 10-15, 5-3, 5-7, 15-20, 3-1, 20-18).<\/li>\n<li><strong>N\u00fat Cha &amp; N\u00fat Con:<\/strong>\n<ul>\n<li>10 l\u00e0 cha c\u1ee7a 5 v\u00e0 15. 5 v\u00e0 15 l\u00e0 con c\u1ee7a 10.<\/li>\n<li>5 l\u00e0 cha c\u1ee7a 3 v\u00e0 7. 3 v\u00e0 7 l\u00e0 con c\u1ee7a 5.<\/li>\n<li>15 l\u00e0 cha c\u1ee7a 20. 20 l\u00e0 con c\u1ee7a 15 (con ph\u1ea3i, kh\u00f4ng c\u00f3 con tr\u00e1i).<\/li>\n<li>3 l\u00e0 cha c\u1ee7a 1. 1 l\u00e0 con c\u1ee7a 3 (con tr\u00e1i, kh\u00f4ng c\u00f3 con ph\u1ea3i).<\/li>\n<li>20 l\u00e0 cha c\u1ee7a 18. 18 l\u00e0 con c\u1ee7a 20 (con tr\u00e1i, kh\u00f4ng c\u00f3 con ph\u1ea3i).<\/li>\n<\/ul>\n<\/li>\n<li><strong>N\u00fat L\u00e1:<\/strong> C\u00e1c n\u00fat kh\u00f4ng c\u00f3 con: 7, 1, 18.<\/li>\n<li><strong>N\u00fat Trung Gian:<\/strong> C\u00e1c n\u00fat c\u00f3 \u00edt nh\u1ea5t m\u1ed9t con: 10, 5, 15, 3, 20.<\/li>\n<li><strong>C\u00e2y Con:<\/strong>\n<ul>\n<li>N\u00fat 5 l\u00e0 g\u1ed1c c\u1ee7a c\u00e2y con tr\u00e1i c\u1ee7a n\u00fat 10 (g\u1ed3m c\u00e1c n\u00fat 5, 3, 7, 1).<\/li>\n<li>N\u00fat 15 l\u00e0 g\u1ed1c c\u1ee7a c\u00e2y con ph\u1ea3i c\u1ee7a n\u00fat 10 (g\u1ed3m c\u00e1c n\u00fat 15, 20, 18).<\/li>\n<\/ul>\n<\/li>\n<li><strong>\u0110\u1ed9 S\u00e2u:<\/strong>\n<ul>\n<li>\u0110\u1ed9 s\u00e2u c\u1ee7a 10 (g\u1ed1c) l\u00e0 0.<\/li>\n<li>\u0110\u1ed9 s\u00e2u c\u1ee7a 5, 15 l\u00e0 1.<\/li>\n<li>\u0110\u1ed9 s\u00e2u c\u1ee7a 3, 7, 20 l\u00e0 2.<\/li>\n<li>\u0110\u1ed9 s\u00e2u c\u1ee7a 1, 18 l\u00e0 3.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Chi\u1ec1u Cao:<\/strong> \u0110\u01b0\u1eddng \u0111i d\u00e0i nh\u1ea5t t\u1eeb g\u1ed1c \u0111\u1ebfn l\u00e1 l\u00e0 10 -&gt; 15 -&gt; 20 -&gt; 18 (ho\u1eb7c 10 -&gt; 5 -&gt; 3 -&gt; 1), c\u00f3 3 c\u1ea1nh. V\u1eady <strong>chi\u1ec1u cao<\/strong> c\u1ee7a c\u00e2y n\u00e0y l\u00e0 3.<\/li>\n<li><strong>B\u1eadc c\u1ee7a N\u00fat:<\/strong>\n<ul>\n<li>B\u1eadc 2: 10, 5.<\/li>\n<li>B\u1eadc 1: 15, 3, 20.<\/li>\n<li>B\u1eadc 0: 7, 1, 18 (c\u00e1c n\u00fat l\u00e1).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>V\u00ed d\u1ee5<\/strong> n\u00e0y cho th\u1ea5y c\u00e1ch c\u00e1c kh\u00e1i ni\u1ec7m li\u00ean k\u1ebft v\u1edbi nhau trong m\u1ed9t c\u1ea5u tr\u00fac c\u00e2y c\u1ee5 th\u1ec3. B\u1ea1n c\u00f3 th\u1ec3 th\u1eed v\u1ebd l\u1ea1i c\u00e2y n\u00e0y ra gi\u1ea5y \u0111\u1ec3 h\u00ecnh dung r\u00f5 h\u01a1n m\u1ed1i quan h\u1ec7 gi\u1eefa c\u00e1c n\u00fat.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Gioi-thieu-mot-so-loai-Cay-Nhi-Phan-pho-bien\"><\/span>Gi\u1edbi thi\u1ec7u m\u1ed9t s\u1ed1 lo\u1ea1i C\u00e2y Nh\u1ecb Ph\u00e2n ph\u1ed5 bi\u1ebfn<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>C\u00e2y nh\u1ecb ph\u00e2n<\/strong> l\u00e0 m\u1ed9t kh\u00e1i ni\u1ec7m t\u1ed5ng qu\u00e1t. Tr\u00ean th\u1ef1c t\u1ebf, c\u00f3 nhi\u1ec1u lo\u1ea1i c\u00e2y nh\u1ecb ph\u00e2n \u0111\u1eb7c bi\u1ec7t \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a d\u1ef1a tr\u00ean c\u00e1c r\u00e0ng bu\u1ed9c ho\u1eb7c t\u00ednh ch\u1ea5t b\u1ed5 sung. Hi\u1ec3u v\u1ec1 ch\u00fang gi\u00fap m\u1edf r\u1ed9ng ki\u1ebfn th\u1ee9c v\u00e0 \u1ee9ng d\u1ee5ng c\u1ee7a b\u1ea1n. D\u01b0\u1edbi \u0111\u00e2y l\u00e0 gi\u1edbi thi\u1ec7u s\u01a1 l\u01b0\u1ee3c v\u1ec1 ba lo\u1ea1i ph\u1ed5 bi\u1ebfn:<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Cay-Nhi-Phan-Tim-Kiem-Binary-Search-Tree-%E2%80%93-BST\"><\/span>C\u00e2y Nh\u1ecb Ph\u00e2n T\u00ecm Ki\u1ebfm (Binary Search Tree &#8211; BST)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0110\u00e2y l\u00e0 lo\u1ea1i c\u00e2y nh\u1ecb ph\u00e2n \u0111\u1eb7c bi\u1ec7t v\u00e0 c\u1ef1c k\u1ef3 quan tr\u1ecdng. <strong>C\u00e2y nh\u1ecb ph\u00e2n t\u00ecm ki\u1ebfm (BST)<\/strong> c\u00f3 th\u00eam m\u1ed9t r\u00e0ng bu\u1ed9c ch\u1eb7t ch\u1ebd v\u1ec1 gi\u00e1 tr\u1ecb d\u1eef li\u1ec7u l\u01b0u tr\u1eef trong c\u00e1c n\u00fat:<\/p>\n<ul>\n<li>V\u1edbi m\u1ecdi n\u00fat trong c\u00e2y, t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb trong <strong>c\u00e2y con tr\u00e1i<\/strong> c\u1ee7a n\u00f3 ph\u1ea3i nh\u1ecf h\u01a1n gi\u00e1 tr\u1ecb c\u1ee7a n\u00fat \u0111\u00f3.<\/li>\n<li>T\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb trong <strong>c\u00e2y con ph\u1ea3i<\/strong> c\u1ee7a n\u00f3 ph\u1ea3i l\u1edbn h\u01a1n (ho\u1eb7c b\u1eb1ng, t\u00f9y quy \u01b0\u1edbc) gi\u00e1 tr\u1ecb c\u1ee7a n\u00fat \u0111\u00f3.<\/li>\n<\/ul>\n<p><strong>\u0110\u1eb7c \u0111i\u1ec3m<\/strong> n\u00e0y gi\u00fap vi\u1ec7c <strong>t\u00ecm ki\u1ebfm<\/strong>, th\u00eam, x\u00f3a ph\u1ea7n t\u1eed trong c\u00e2y tr\u1edf n\u00ean r\u1ea5t hi\u1ec7u qu\u1ea3 (trung b\u00ecnh l\u00e0 O(log N) v\u1edbi c\u00e2y c\u00e2n b\u1eb1ng). <strong>BST<\/strong> l\u00e0 n\u1ec1n t\u1ea3ng cho nhi\u1ec1u <strong>c\u1ea5u tr\u00fac d\u1eef li\u1ec7u<\/strong> v\u00e0 <strong>thu\u1eadt to\u00e1n<\/strong> kh\u00e1c.<\/p>\n<p>V\u00ed d\u1ee5 c\u00e2y \u1edf ph\u1ea7n tr\u01b0\u1edbc <i>kh\u00f4ng ph\u1ea3i<\/i> l\u00e0 BST v\u00ec v\u00ed d\u1ee5, n\u00fat 18 (trong c\u00e2y con ph\u1ea3i c\u1ee7a 15) l\u1ea1i nh\u1ecf h\u01a1n 20 (cha tr\u1ef1c ti\u1ebfp). M\u1ed9t v\u00ed d\u1ee5 v\u1ec1 BST:<\/p>\n<pre><code class=\"language-plaintext\">      8\r\n     \/ \\\r\n    3   10\r\n   \/ \\    \\\r\n  1   6    14\r\n     \/ \\  \/\r\n    4   7 13\r\n<\/code><\/pre>\n<p>B\u1ea1n c\u00f3 th\u1ec3 ki\u1ec3m tra th\u1ea5y m\u1ecdi n\u00fat \u0111\u1ec1u tu\u00e2n th\u1ee7 quy t\u1eafc gi\u00e1 tr\u1ecb c\u1ee7a BST.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Cay-Nhi-Phan-Day-Du-Full-Binary-Tree\"><\/span>C\u00e2y Nh\u1ecb Ph\u00e2n \u0110\u1ea7y \u0110\u1ee7 (Full Binary Tree)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>C\u00e2y nh\u1ecb ph\u00e2n \u0111\u1ea7y \u0111\u1ee7<\/strong> (c\u00f2n g\u1ecdi l\u00e0 <strong>Proper Binary Tree<\/strong>) l\u00e0 c\u00e2y m\u00e0 m\u1ed7i <strong>n\u00fat<\/strong> trong \u0111\u00f3 ch\u1ec9 c\u00f3 th\u1ec3 c\u00f3 <strong>0 ho\u1eb7c 2 n\u00fat con<\/strong>. Kh\u00f4ng c\u00f3 n\u00fat n\u00e0o \u0111\u01b0\u1ee3c ph\u00e9p ch\u1ec9 c\u00f3 1 n\u00fat con.<\/p>\n<p>N\u00f3i c\u00e1ch kh\u00e1c, m\u1ecdi n\u00fat trong <strong>c\u00e2y nh\u1ecb ph\u00e2n \u0111\u1ea7y \u0111\u1ee7<\/strong> ho\u1eb7c l\u00e0 <strong>n\u00fat l\u00e1<\/strong> (0 con) ho\u1eb7c l\u00e0 <strong>n\u00fat trung gian<\/strong> c\u00f3 \u0111\u1ee7 c\u1ea3 hai con (tr\u00e1i v\u00e0 ph\u1ea3i).<\/p>\n<p><strong>V\u00ed d\u1ee5<\/strong> v\u1ec1 c\u00e2y \u0111\u1ea7y \u0111\u1ee7:<\/p>\n<pre><code class=\"language-plaintext\">      A\r\n     \/ \\\r\n    B   C\r\n       \/ \\\r\n      D   E\r\n<\/code><\/pre>\n<p>Trong c\u00e2y n\u00e0y, A, C l\u00e0 n\u00fat c\u00f3 2 con; B, D, E l\u00e0 n\u00fat l\u00e1 (0 con). Kh\u00f4ng c\u00f3 n\u00fat n\u00e0o ch\u1ec9 c\u00f3 1 con. C\u00e2y trong v\u00ed d\u1ee5 minh h\u1ecda \u0111\u1ea7u ti\u00ean <i>kh\u00f4ng ph\u1ea3i<\/i> l\u00e0 c\u00e2y \u0111\u1ea7y \u0111\u1ee7 (v\u00ed d\u1ee5 n\u00fat 15, 3, 20 ch\u1ec9 c\u00f3 1 con).<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Cay-Nhi-Phan-Hoan-Chinh-Complete-Binary-Tree\"><\/span>C\u00e2y Nh\u1ecb Ph\u00e2n Ho\u00e0n Ch\u1ec9nh (Complete Binary Tree)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>C\u00e2y nh\u1ecb ph\u00e2n ho\u00e0n ch\u1ec9nh<\/strong> l\u00e0 lo\u1ea1i c\u00e2y c\u00f3 m\u1ed9t \u0111\u1ecbnh ngh\u0129a h\u01a1i kh\u00e1c m\u1ed9t ch\u00fat. M\u1ed9t c\u00e2y nh\u1ecb ph\u00e2n \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 <strong>ho\u00e0n ch\u1ec9nh<\/strong> n\u1ebfu:<\/p>\n<ol>\n<li>T\u1ea5t c\u1ea3 c\u00e1c m\u1ee9c, ngo\u1ea1i tr\u1eeb c\u00f3 th\u1ec3 l\u00e0 m\u1ee9c cu\u1ed1i c\u00f9ng, \u0111\u1ec1u \u0111\u01b0\u1ee3c l\u1ea5p \u0111\u1ea7y ho\u00e0n to\u00e0n.<\/li>\n<li>N\u1ebfu m\u1ee9c cu\u1ed1i c\u00f9ng ch\u01b0a \u0111\u1ea7y, th\u00ec c\u00e1c n\u00fat \u1edf m\u1ee9c n\u00e0y ph\u1ea3i \u0111\u01b0\u1ee3c \u0111i\u1ec1n t\u1eeb tr\u00e1i sang ph\u1ea3i.<\/li>\n<\/ol>\n<p><strong>\u0110\u1eb7c \u0111i\u1ec3m<\/strong> n\u00e0y l\u00e0m cho <strong>c\u00e2y nh\u1ecb ph\u00e2n ho\u00e0n ch\u1ec9nh<\/strong> r\u1ea5t ph\u00f9 h\u1ee3p \u0111\u1ec3 bi\u1ec3u di\u1ec5n b\u1eb1ng m\u1ea3ng m\u1ed9t c\u00e1ch hi\u1ec7u qu\u1ea3, \u0111\u1eb7c bi\u1ec7t trong c\u1ea5u tr\u00fac d\u1eef li\u1ec7u <strong>Heap<\/strong> (\u0111\u1ed1ng).<\/p>\n<p><strong>V\u00ed d\u1ee5<\/strong> v\u1ec1 c\u00e2y ho\u00e0n ch\u1ec9nh:<\/p>\n<pre><code class=\"language-plaintext\">      1\r\n     \/ \\\r\n    2   3\r\n   \/ \\ \/\r\n  4  5 6\r\n<\/code><\/pre>\n<p>C\u00e2y n\u00e0y c\u00f3 chi\u1ec1u cao 2. M\u1ee9c 0 (n\u00fat 1), m\u1ee9c 1 (n\u00fat 2, 3) \u0111\u1ec1u \u0111\u1ea7y. M\u1ee9c 2 (m\u1ee9c cu\u1ed1i) c\u00f3 n\u00fat 4, 5, 6 \u0111\u01b0\u1ee3c \u0111i\u1ec1n t\u1eeb tr\u00e1i sang ph\u1ea3i. N\u1ebfu th\u00eam n\u00fat 7, n\u00f3 ph\u1ea3i l\u00e0 con ph\u1ea3i c\u1ee7a n\u00fat 3.<\/p>\n<p>C\u1ea7n ph\u00e2n bi\u1ec7t r\u00f5 r\u00e0ng gi\u1eefa c\u00e2y <strong>\u0111\u1ea7y \u0111\u1ee7<\/strong> v\u00e0 c\u00e2y <strong>ho\u00e0n ch\u1ec9nh<\/strong>. M\u1ed9t c\u00e2y c\u00f3 th\u1ec3 l\u00e0 \u0111\u1ea7y \u0111\u1ee7 nh\u01b0ng kh\u00f4ng ho\u00e0n ch\u1ec9nh, ho\u00e0n ch\u1ec9nh nh\u01b0ng kh\u00f4ng \u0111\u1ea7y \u0111\u1ee7, v\u1eeba \u0111\u1ea7y \u0111\u1ee7 v\u1eeba ho\u00e0n ch\u1ec9nh, ho\u1eb7c kh\u00f4ng l\u00e0 c\u1ea3 hai.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan-02.jpg\" alt=\"C\u00e2y nh\u1ecb ph\u00e2n 02\" width=\"750\" height=\"452\" class=\"aligncenter size-full wp-image-27388\" title=\"\" srcset=\"https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan-02.jpg 750w, https:\/\/interdata.vn\/blog\/wp-content\/uploads\/2025\/04\/Cay-nhi-phan-02-300x181.jpg 300w\" sizes=\"auto, (max-width: 750px) 100vw, 750px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Ung-dung-co-ban-cua-Cay-Nhi-Phan-trong-thuc-te\"><\/span>\u1ee8ng d\u1ee5ng c\u01a1 b\u1ea3n c\u1ee7a C\u00e2y Nh\u1ecb Ph\u00e2n trong th\u1ef1c t\u1ebf<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>V\u1eady c\u00e2u h\u1ecfi \u0111\u1eb7t ra l\u00e0, h\u1ecdc <strong>c\u00e2y nh\u1ecb ph\u00e2n \u0111\u1ec3 l\u00e0m g\u00ec<\/strong>? <strong>C\u1ea5u tr\u00fac d\u1eef li\u1ec7u<\/strong> n\u00e0y c\u00f3 v\u1ebb tr\u1eebu t\u01b0\u1ee3ng, nh\u01b0ng n\u00f3 c\u00f3 r\u1ea5t nhi\u1ec1u <strong>\u1ee9ng d\u1ee5ng<\/strong> quan tr\u1ecdng v\u00e0 l\u00e0 n\u1ec1n t\u1ea3ng trong <strong>l\u1eadp tr\u00ecnh<\/strong>:<\/p>\n<ol>\n<li><strong>Bi\u1ec3u di\u1ec5n c\u1ea5u tr\u00fac ph\u00e2n c\u1ea5p:<\/strong> C\u00e2y nh\u1ecb ph\u00e2n r\u1ea5t t\u1ef1 nhi\u00ean \u0111\u1ec3 m\u00f4 t\u1ea3 c\u00e1c m\u1ed1i quan h\u1ec7 ph\u00e2n c\u1ea5p. V\u00ed d\u1ee5 nh\u01b0 s\u01a1 \u0111\u1ed3 t\u1ed5 ch\u1ee9c c\u00f4ng ty, h\u1ec7 th\u1ed1ng ph\u00e2n lo\u1ea1i sinh h\u1ecdc, hay c\u1ea5u tr\u00fac <strong>DOM (Document Object Model)<\/strong> c\u1ee7a m\u1ed9t trang web HTML.<\/li>\n<li><strong>C\u00e2y Nh\u1ecb Ph\u00e2n T\u00ecm Ki\u1ebfm (BST):<\/strong> Nh\u01b0 \u0111\u00e3 \u0111\u1ec1 c\u1eadp, BST cho ph\u00e9p <strong>t\u00ecm ki\u1ebfm<\/strong>, th\u00eam, x\u00f3a d\u1eef li\u1ec7u r\u1ea5t nhanh ch\u00f3ng. N\u00f3 \u0111\u01b0\u1ee3c d\u00f9ng trong vi\u1ec7c c\u00e0i \u0111\u1eb7t c\u00e1c c\u1ea5u tr\u00fac nh\u01b0 t\u1eadp h\u1ee3p (Set) v\u00e0 \u00e1nh x\u1ea1 (Map) trong nhi\u1ec1u th\u01b0 vi\u1ec7n chu\u1ea9n.<\/li>\n<li><strong>C\u00e2y bi\u1ec3u th\u1ee9c (Expression Trees):<\/strong> C\u00e2y nh\u1ecb ph\u00e2n c\u00f3 th\u1ec3 d\u00f9ng \u0111\u1ec3 bi\u1ec3u di\u1ec5n c\u00e1c bi\u1ec3u th\u1ee9c to\u00e1n h\u1ecdc. C\u00e1c to\u00e1n t\u1eed (+, -, *, \/) l\u00e0 c\u00e1c n\u00fat trung gian, c\u00f2n c\u00e1c to\u00e1n h\u1ea1ng (s\u1ed1) l\u00e0 c\u00e1c n\u00fat l\u00e1. Vi\u1ec7c duy\u1ec7t c\u00e2y theo c\u00e1c th\u1ee9 t\u1ef1 kh\u00e1c nhau (inorder, preorder, postorder) s\u1ebd cho ra c\u00e1c d\u1ea1ng bi\u1ec3u di\u1ec5n kh\u00e1c nhau c\u1ee7a bi\u1ec3u th\u1ee9c (trung t\u1ed1, ti\u1ec1n t\u1ed1, h\u1eadu t\u1ed1).<\/li>\n<li><strong>C\u1ea5u tr\u00fac Heap (\u0110\u1ed1ng):<\/strong> Heap l\u00e0 m\u1ed9t c\u1ea5u tr\u00fac d\u1eef li\u1ec7u d\u1ef1a tr\u00ean <strong>c\u00e2y nh\u1ecb ph\u00e2n ho\u00e0n ch\u1ec9nh<\/strong>, th\u01b0\u1eddng \u0111\u01b0\u1ee3c d\u00f9ng \u0111\u1ec3 c\u00e0i \u0111\u1eb7t h\u00e0ng \u0111\u1ee3i \u01b0u ti\u00ean (Priority Queue). Heap r\u1ea5t hi\u1ec7u qu\u1ea3 trong vi\u1ec7c t\u00ecm ph\u1ea7n t\u1eed l\u1edbn nh\u1ea5t\/nh\u1ecf nh\u1ea5t.<\/li>\n<li><strong>Thu\u1eadt to\u00e1n n\u00e9n d\u1eef li\u1ec7u:<\/strong> C\u00e1c thu\u1eadt to\u00e1n n\u00e9n nh\u01b0 Huffman <a href=\"https:\/\/interdata.vn\/blog\/coding-la-gi\/\">Coding<\/a> s\u1eed d\u1ee5ng c\u00e2y nh\u1ecb ph\u00e2n \u0111\u1ec3 x\u00e2y d\u1ef1ng m\u00e3 t\u1ed1i \u01b0u cho c\u00e1c k\u00fd t\u1ef1 d\u1ef1a tr\u00ean t\u1ea7n su\u1ea5t xu\u1ea5t hi\u1ec7n c\u1ee7a ch\u00fang, gi\u00fap gi\u1ea3m k\u00edch th\u01b0\u1edbc d\u1eef li\u1ec7u hi\u1ec7u qu\u1ea3.<\/li>\n<li><strong>C\u00e2y quy\u1ebft \u0111\u1ecbnh (Decision Trees):<\/strong> Trong l\u0129nh v\u1ef1c h\u1ecdc m\u00e1y (Machine Learning), c\u00e2y quy\u1ebft \u0111\u1ecbnh l\u00e0 m\u1ed9t m\u00f4 h\u00ecnh d\u1ef1 \u0111o\u00e1n s\u1eed d\u1ee5ng c\u1ea5u tr\u00fac c\u00e2y (th\u01b0\u1eddng l\u00e0 c\u00e2y nh\u1ecb ph\u00e2n) \u0111\u1ec3 \u0111\u01b0a ra quy\u1ebft \u0111\u1ecbnh d\u1ef1a tr\u00ean c\u00e1c \u0111\u1eb7c tr\u01b0ng c\u1ee7a d\u1eef li\u1ec7u.<\/li>\n<\/ol>\n<p>Nh\u1eefng <strong>\u1ee9ng d\u1ee5ng<\/strong> n\u00e0y cho th\u1ea5y t\u1ea7m quan tr\u1ecdng c\u1ee7a vi\u1ec7c n\u1eafm v\u1eefng <strong>kh\u00e1i ni\u1ec7m c\u00e2y nh\u1ecb ph\u00e2n<\/strong>. N\u00f3 kh\u00f4ng ch\u1ec9 l\u00e0 l\u00fd thuy\u1ebft su\u00f4ng m\u00e0 c\u00f2n l\u00e0 c\u00f4ng c\u1ee5 m\u1ea1nh m\u1ebd \u0111\u01b0\u1ee3c \u00e1p d\u1ee5ng trong nhi\u1ec1u b\u00e0i to\u00e1n th\u1ef1c t\u1ebf.<\/p>\n<div style=\"background-color: #e6f2ff; border-radius: 10px; padding: 20px; margin: 20px 0; border: 1px solid #b3d9ff;\">\n<p>Khi b\u1ea1n \u0111\u00e3 n\u1eafm v\u1eefng c\u00e1c kh\u00e1i ni\u1ec7m nh\u01b0 c\u00e2y nh\u1ecb ph\u00e2n v\u00e0 b\u1eaft \u0111\u1ea7u \u00e1p d\u1ee5ng ch\u00fang v\u00e0o vi\u1ec7c x\u00e2y d\u1ef1ng website hay \u1ee9ng d\u1ee5ng th\u1ef1c t\u1ebf, vi\u1ec7c l\u1ef1a ch\u1ecdn m\u1ed9t n\u1ec1n t\u1ea3ng l\u01b0u tr\u1eef \u1ed5n \u0111\u1ecbnh l\u00e0 r\u1ea5t quan tr\u1ecdng. B\u1ea1n c\u00f3 th\u1ec3 b\u1eaft \u0111\u1ea7u v\u1edbi d\u1ecbch v\u1ee5 <a target=\"_blank\" rel=\"noopener noreferrer\" href=\"https:\/\/interdata.vn\/thue-hosting\/\"><strong>thu\u00ea Hosting<\/strong><\/a> t\u1ea1i InterData, v\u1eadn h\u00e0nh tr\u00ean ph\u1ea7n c\u1ee9ng chuy\u00ean d\u1ee5ng th\u1ebf h\u1ec7 m\u1edbi, t\u1ed1i \u01b0u dung l\u01b0\u1ee3ng.<\/p>\n<p>N\u1ebfu d\u1ef1 \u00e1n c\u1ee7a b\u1ea1n \u0111\u00f2i h\u1ecfi nhi\u1ec1u t\u00e0i nguy\u00ean v\u00e0 kh\u1ea3 n\u0103ng ki\u1ec3m so\u00e1t cao h\u01a1n, h\u00e3y c\u00e2n nh\u1eafc d\u1ecbch v\u1ee5 <a target=\"_blank\" rel=\"noopener noreferrer\" href=\"https:\/\/interdata.vn\/thue-vps\/\"><strong>thu\u00ea VPS<\/strong><\/a> m\u1ea1nh m\u1ebd. Cho c\u00e1c d\u1ef1 \u00e1n quy m\u00f4 l\u1edbn c\u1ea7n linh ho\u1ea1t t\u1ed1i \u0111a, kh\u00e1m ph\u00e1 d\u1ecbch v\u1ee5 <a target=\"_blank\" rel=\"noopener noreferrer\" href=\"https:\/\/interdata.vn\/cloud-server\/\"><strong>thu\u00ea Cloud Server<\/strong><\/a> t\u1ea1i InterData.<\/p>\n<p>M\u1ecdi gi\u1ea3i ph\u00e1p l\u01b0u tr\u1eef \u1edf \u0111\u00e2y \u0111\u1ec1u d\u00f9ng ph\u1ea7n c\u1ee9ng cao c\u1ea5p nh\u01b0 b\u1ed9 x\u1eed l\u00fd AMD EPYC Gen 3th, \u1ed5 c\u1ee9ng SSD NVMe U.2 t\u1ed1c \u0111\u1ed9 cao, mang l\u1ea1i s\u1ef1 \u1ed5n \u0111\u1ecbnh, b\u0103ng th\u00f4ng l\u1edbn v\u1edbi m\u1ee9c gi\u00e1 h\u1ee3p l\u00fd nh\u1edd c\u00f4ng ngh\u1ec7 \u1ea3o h\u00f3a ti\u00ean ti\u1ebfn.<\/p>\n<\/div>\n<h2><span class=\"ez-toc-section\" id=\"Nhung-nham-lan-thuong-gap-khi-hoc-Cay-Nhi-Phan\"><\/span>Nh\u1eefng nh\u1ea7m l\u1eabn th\u01b0\u1eddng g\u1eb7p khi h\u1ecdc C\u00e2y Nh\u1ecb Ph\u00e2n<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Khi m\u1edbi ti\u1ebfp c\u1eadn <strong>c\u00e2y nh\u1ecb ph\u00e2n<\/strong>, ng\u01b0\u1eddi h\u1ecdc th\u01b0\u1eddng g\u1eb7p m\u1ed9t s\u1ed1 \u0111i\u1ec3m d\u1ec5 g\u00e2y nh\u1ea7m l\u1eabn. Hi\u1ec3u r\u00f5 nh\u1eefng \u0111i\u1ec1u n\u00e0y s\u1ebd gi\u00fap b\u1ea1n tr\u00e1nh \u0111\u01b0\u1ee3c c\u00e1c sai s\u00f3t kh\u00f4ng \u0111\u00e1ng c\u00f3:<\/p>\n<ol>\n<li><strong>Nh\u1ea7m l\u1eabn Chi\u1ec1u cao v\u00e0 \u0110\u1ed9 s\u00e2u:<\/strong> <strong>\u0110\u1ed9 s\u00e2u<\/strong> l\u00e0 thu\u1ed9c t\u00ednh c\u1ee7a <i>t\u1eebng n\u00fat<\/i> (kho\u1ea3ng c\u00e1ch t\u1eeb g\u1ed1c \u0111\u1ebfn n\u00fat \u0111\u00f3), trong khi <strong>Chi\u1ec1u cao<\/strong> l\u00e0 thu\u1ed9c t\u00ednh c\u1ee7a <i>to\u00e0n b\u1ed9 c\u00e2y<\/i> (\u0111\u1ed9 s\u00e2u l\u1edbn nh\u1ea5t c\u1ee7a n\u00fat l\u00e1). Chi\u1ec1u cao c\u1ee7a c\u00e2y b\u1eb1ng chi\u1ec1u cao c\u1ee7a n\u00fat g\u1ed1c.<\/li>\n<li><strong>C\u00e2y Nh\u1ecb Ph\u00e2n vs. C\u00e2y Nh\u1ecb Ph\u00e2n T\u00ecm Ki\u1ebfm (BST):<\/strong> Kh\u00f4ng ph\u1ea3i m\u1ecdi c\u00e2y nh\u1ecb ph\u00e2n \u0111\u1ec1u l\u00e0 BST. BST l\u00e0 m\u1ed9t tr\u01b0\u1eddng h\u1ee3p <i>\u0111\u1eb7c bi\u1ec7t<\/i> c\u1ee7a c\u00e2y nh\u1ecb ph\u00e2n c\u00f3 th\u00eam r\u00e0ng bu\u1ed9c v\u1ec1 gi\u00e1 tr\u1ecb n\u00fat. C\u1ea7n ki\u1ec3m tra \u0111i\u1ec1u ki\u1ec7n gi\u00e1 tr\u1ecb m\u1edbi k\u1ebft lu\u1eadn m\u1ed9t c\u00e2y l\u00e0 BST.<\/li>\n<li><strong>C\u00e2y \u0110\u1ea7y \u0111\u1ee7 vs. C\u00e2y Ho\u00e0n ch\u1ec9nh:<\/strong> Hai kh\u00e1i ni\u1ec7m n\u00e0y kh\u00e1c nhau. C\u00e2y <strong>\u0111\u1ea7y \u0111\u1ee7<\/strong> y\u00eau c\u1ea7u m\u1ecdi n\u00fat ph\u1ea3i c\u00f3 0 ho\u1eb7c 2 con. C\u00e2y <strong>ho\u00e0n ch\u1ec9nh<\/strong> y\u00eau c\u1ea7u c\u00e1c m\u1ee9c ph\u1ea3i \u0111\u1ea7y t\u1eeb tr\u00e1i sang, tr\u1eeb m\u1ee9c cu\u1ed1i c\u00f9ng. H\u00e3y xem l\u1ea1i \u0111\u1ecbnh ngh\u0129a v\u00e0 v\u00ed d\u1ee5 \u0111\u1ec3 ph\u00e2n bi\u1ec7t r\u00f5.<\/li>\n<li><strong>Th\u1ee9 t\u1ef1 con:<\/strong> Trong c\u00e2y nh\u1ecb ph\u00e2n, vi\u1ec7c m\u1ed9t n\u00fat l\u00e0 con tr\u00e1i hay con ph\u1ea3i r\u1ea5t quan tr\u1ecdng v\u00e0 t\u1ea1o ra c\u00e1c c\u00e2y kh\u00e1c nhau v\u1ec1 m\u1eb7t c\u1ea5u tr\u00fac, ngay c\u1ea3 khi ch\u00fang ch\u1ee9a c\u00f9ng m\u1ed9t t\u1eadp h\u1ee3p c\u00e1c n\u00fat.<\/li>\n<li><strong>Bi\u1ec3u di\u1ec5n b\u1eb1ng m\u1ea3ng:<\/strong> Ch\u1ec9 c\u00f3 <strong>c\u00e2y nh\u1ecb ph\u00e2n ho\u00e0n ch\u1ec9nh<\/strong> m\u1edbi th\u1ef1c s\u1ef1 ph\u00f9 h\u1ee3p \u0111\u1ec3 bi\u1ec3u di\u1ec5n hi\u1ec7u qu\u1ea3 b\u1eb1ng m\u1ea3ng. Vi\u1ec7c c\u1ed1 g\u1eafng bi\u1ec3u di\u1ec5n c\u00e2y kh\u00f4ng ho\u00e0n ch\u1ec9nh b\u1eb1ng m\u1ea3ng c\u00f3 th\u1ec3 g\u00e2y l\u00e3ng ph\u00ed b\u1ed9 nh\u1edb l\u1edbn.<\/li>\n<\/ol>\n<p>Nh\u1eadn bi\u1ebft v\u00e0 tr\u00e1nh \u0111\u01b0\u1ee3c nh\u1eefng nh\u1ea7m l\u1eabn n\u00e0y s\u1ebd gi\u00fap b\u1ea1n x\u00e2y d\u1ef1ng n\u1ec1n t\u1ea3ng ki\u1ebfn th\u1ee9c v\u1eefng ch\u1eafc h\u01a1n v\u1ec1 <strong>c\u1ea5u tr\u00fac d\u1eef li\u1ec7u<\/strong> quan tr\u1ecdng n\u00e0y.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>N\u1ebfu b\u1ea1n \u0111ang h\u1ecdc c\u1ea5u tr\u00fac d\u1eef li\u1ec7u v\u00e0 thu\u1eadt to\u00e1n, th\u00ec kh\u00e1i ni\u1ec7m c\u00e2y nh\u1ecb ph\u00e2n (binary tree) ch\u1eafc ch\u1eafn l\u00e0 ch\u1ee7 \u0111\u1ec1 kh\u00f4ng th\u1ec3 b\u1ecf qua. V\u1edbi thi\u1ebft k\u1ebf \u0111\u01a1n gi\u1ea3n nh\u01b0ng c\u1ef1c k\u1ef3 hi\u1ec7u qu\u1ea3, c\u00e2y nh\u1ecb ph\u00e2n xu\u1ea5t hi\u1ec7n \u1edf m\u1ecdi n\u01a1i trong l\u1eadp tr\u00ecnh. Trong b\u00e0i vi\u1ebft n\u00e0y, b\u1ea1n s\u1ebd<\/p>\n","protected":false},"author":2,"featured_media":27390,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[140],"tags":[],"class_list":["post-27385","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-lap-trinh"],"_links":{"self":[{"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/posts\/27385","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/comments?post=27385"}],"version-history":[{"count":2,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/posts\/27385\/revisions"}],"predecessor-version":[{"id":27391,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/posts\/27385\/revisions\/27391"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/media\/27390"}],"wp:attachment":[{"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/media?parent=27385"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/categories?post=27385"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/interdata.vn\/blog\/wp-json\/wp\/v2\/tags?post=27385"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}