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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Types_of_trees" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_trees"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Types of trees</span> </div> </a> <button aria-controls="toc-Types_of_trees-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Types of trees subsection</span> </button> <ul id="toc-Types_of_trees-sublist" class="vector-toc-list"> <li id="toc-Binary_search_tree" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Binary_search_tree"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Binary search tree</span> </div> </a> <ul id="toc-Binary_search_tree-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-B-tree" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#B-tree"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>B-tree</span> </div> </a> <ul id="toc-B-tree-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-(a,b)-tree" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#(a,b)-tree"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>(a,b)-tree</span> </div> </a> <ul id="toc-(a,b)-tree-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ternary_search_tree" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ternary_search_tree"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Ternary search tree</span> </div> </a> <ul id="toc-Ternary_search_tree-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Searching_algorithms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Searching_algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Searching algorithms</span> </div> </a> <button aria-controls="toc-Searching_algorithms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Searching algorithms subsection</span> </button> <ul id="toc-Searching_algorithms-sublist" class="vector-toc-list"> <li id="toc-Searching_for_a_specific_key" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Searching_for_a_specific_key"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Searching for a specific key</span> </div> </a> <ul id="toc-Searching_for_a_specific_key-sublist" class="vector-toc-list"> <li id="toc-Recursive" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Recursive"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Recursive</span> </div> </a> <ul id="toc-Recursive-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Iterative" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Iterative"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Iterative</span> </div> </a> <ul id="toc-Iterative-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Searching_for_min_and_max" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Searching_for_min_and_max"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Searching for min and max</span> </div> </a> <ul id="toc-Searching_for_min_and_max-sublist" class="vector-toc-list"> <li id="toc-Minimum" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Minimum"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Minimum</span> </div> </a> <ul id="toc-Minimum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maximum" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Maximum"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Maximum</span> </div> </a> <ul id="toc-Maximum-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" 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href="https://de.wikipedia.org/wiki/Suchbaum" title="Suchbaum – German" lang="de" hreflang="de" data-title="Suchbaum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81rbol_de_b%C3%BAsqueda" title="Árbol de búsqueda – Spanish" lang="es" hreflang="es" data-title="Árbol de búsqueda" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%B3%D8%AA%D8%AC%D9%88%DB%8C_%D8%AF%D8%B1%D8%AE%D8%AA%DB%8C" title="جستجوی درختی – Persian" lang="fa" hreflang="fa" data-title="جستجوی درختی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Arbre_de_recherche" title="Arbre de recherche – French" lang="fr" hreflang="fr" data-title="Arbre de recherche" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%83%90%EC%83%89_%ED%8A%B8%EB%A6%AC" title="탐색 트리 – Korean" lang="ko" hreflang="ko" data-title="탐색 트리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Zoekboom" title="Zoekboom – Dutch" lang="nl" hreflang="nl" data-title="Zoekboom" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%8E%A2%E7%B4%A2%E6%9C%A8" title="探索木 – Japanese" lang="ja" hreflang="ja" data-title="探索木" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81rvore_de_busca" title="Árvore de busca – Portuguese" lang="pt" hreflang="pt" data-title="Árvore de busca" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Pretraga_stabla" title="Pretraga stabla – Serbian" lang="sr" hreflang="sr" data-title="Pretraga stabla" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hakupuu" title="Hakupuu – Finnish" lang="fi" hreflang="fi" data-title="Hakupuu" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B5%D1%80%D0%B5%D0%B2%D0%BE_%D0%BF%D0%BE%D1%88%D1%83%D0%BA%D1%83" title="Дерево пошуку – Ukrainian" lang="uk" hreflang="uk" data-title="Дерево пошуку" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%90%9C%E7%B4%A2%E6%A0%91" title="搜索树 – Chinese" lang="zh" hreflang="zh" data-title="搜索树" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Data structure in tree form sorted for fast lookup</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Tree_search" class="mw-redirect" title="Tree search">tree search</a>.</div> <p>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, a <b>search tree</b> is a <a href="/wiki/Tree_(data_structure)" class="mw-redirect" title="Tree (data structure)">tree data structure</a> used for locating specific keys from within a set. In order for a tree to function as a search tree, the key for each node must be greater than any keys in subtrees on the left, and less than any keys in subtrees on the right.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The advantage of search trees is their efficient search time given the tree is reasonably balanced, which is to say the <a href="/wiki/Tree_data_structure#Terminologies_used_in_Trees" class="mw-redirect" title="Tree data structure">leaves</a> at either end are of comparable depths. Various search-tree data structures exist, several of which also allow efficient insertion and deletion of elements, which operations then have to maintain tree balance. </p><p>Search trees are often used to implement an <a href="/wiki/Associative_array" title="Associative array">associative array</a>. The search tree algorithm uses the key from the <a href="/wiki/Attribute%E2%80%93value_pair" class="mw-redirect" title="Attribute–value pair">key–value pair</a> to find a location, and then the application stores the entire key–value pair at that particular location. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Types_of_trees">Types of trees</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=1" title="Edit section: Types of trees"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Binary_search_tree.svg" class="mw-file-description"><img alt="Binary search tree" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Binary_search_tree.svg/220px-Binary_search_tree.svg.png" decoding="async" width="220" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Binary_search_tree.svg/330px-Binary_search_tree.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Binary_search_tree.svg/440px-Binary_search_tree.svg.png 2x" data-file-width="300" data-file-height="250" /></a><figcaption>Binary search tree</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Binary_search_tree">Binary search tree</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=2" title="Edit section: Binary search tree"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Binary_search_tree" title="Binary search tree">Binary search tree</a></div> <p>A Binary Search Tree is a node-based data structure where each node contains a key and two subtrees, the left and right. For all nodes, the left subtree's key must be less than the node's key, and the right subtree's key must be greater than the node's key. These subtrees must all qualify as binary search trees. </p><p>The worst-case <a href="/wiki/Time_complexity" title="Time complexity">time complexity</a> for searching a binary search tree is the <a href="/wiki/Tree_(data_structure)#Terminology_used_in_trees" class="mw-redirect" title="Tree (data structure)">height of the tree</a>, which can be as small as O(log n) for a tree with n elements. </p> <div class="mw-heading mw-heading3"><h3 id="B-tree">B-tree</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=3" title="Edit section: B-tree"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/B-tree" title="B-tree">B-tree</a></div> <p>B-trees are generalizations of binary search trees in that they can have a variable number of subtrees at each node. While child-nodes have a pre-defined range, they will not necessarily be filled with data, meaning B-trees can potentially waste some space. The advantage is that B-trees do not need to be re-balanced as frequently as other <a href="/wiki/Self-balancing_binary_search_tree" title="Self-balancing binary search tree">self-balancing trees</a>. </p><p>Due to the variable range of their node length, B-trees are optimized for systems that read large blocks of data, they are also commonly used in databases. </p><p>The time complexity for searching a B-tree is O(log n). </p> <div class="mw-heading mw-heading3"><h3 id="(a,b)-tree"><span id=".28a.2Cb.29-tree"></span>(a,b)-tree</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=4" title="Edit section: (a,b)-tree"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/(a,b)-tree" class="mw-redirect" title="(a,b)-tree">(a,b)-tree</a></div> <p>An (a,b)-tree is a search tree where all of its leaves are the same depth. Each node has at least <b>a</b> children and at most <b>b</b> children, while the root has at least 2 children and at most <b>b</b> children. </p><p><b>a</b> and <b>b</b> can be decided with the following formula:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\leq a\leq {\frac {(b+1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>≤</mo> <mi>a</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\leq a\leq {\frac {(b+1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48c13c821a07827e3972d2b9c4cdc9ed26993b23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.235ex; height:5.676ex;" alt="{\displaystyle 2\leq a\leq {\frac {(b+1)}{2}}}" /></span> </p><p>The time complexity for searching an (a,b)-tree is O(log n). </p> <div class="mw-heading mw-heading3"><h3 id="Ternary_search_tree">Ternary search tree</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=5" title="Edit section: Ternary search tree"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ternary_search_tree" title="Ternary search tree">Ternary search tree</a></div> <p>A ternary search tree is a type of <a href="/wiki/Tree_(data_structure)" class="mw-redirect" title="Tree (data structure)">tree</a> that can have 3 nodes: a low child, an equal child, and a high child. Each node stores a single character and the tree itself is ordered the same way a binary search tree is, with the exception of a possible third node. </p><p>Searching a ternary search tree involves passing in a <a href="/wiki/String_(computer_science)" title="String (computer science)">string</a> to test whether any path contains it. </p><p>The time complexity for searching a balanced ternary search tree is O(log n). </p> <div class="mw-heading mw-heading2"><h2 id="Searching_algorithms">Searching algorithms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=6" title="Edit section: Searching algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Searching_for_a_specific_key">Searching for a specific key</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=7" title="Edit section: Searching for a specific key"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Assuming the tree is ordered, we can take a key and attempt to locate it within the tree. The following algorithms are generalized for binary search trees, but the same idea can be applied to trees of other formats. </p> <div class="mw-heading mw-heading4"><h4 id="Recursive">Recursive</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=8" title="Edit section: Recursive"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <pre>search-recursive(key, node) <b>if</b> node is <i>NULL</i> <b>return</b> <i>EMPTY_TREE</i> <b>if</b> key < node.key return search-recursive(key, node.left) <b>else if</b> key > node.key return search-recursive(key, node.right) <b>else</b> <b>return</b> node </pre> <div class="mw-heading mw-heading4"><h4 id="Iterative">Iterative</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=9" title="Edit section: Iterative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <pre>searchIterative(key, node) currentNode := node <b>while</b> currentNode is not <i>NULL</i> <b>if</b> currentNode.key = key <b>return</b> currentNode <b>else if</b> currentNode.key > key currentNode := currentNode.left <b>else</b> currentNode := currentNode.right </pre> <div class="mw-heading mw-heading3"><h3 id="Searching_for_min_and_max">Searching for min and max</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=10" title="Edit section: Searching for min and max"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a sorted tree, the minimum is located at the node farthest left, while the maximum is located at the node farthest right.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Minimum">Minimum</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=11" title="Edit section: Minimum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <pre>findMinimum(node) <b>if</b> node is <i>NULL</i> <b>return</b> <i>EMPTY_TREE</i> min := node <b>while</b> min.left is not <i>NULL</i> min := min.left <b>return</b> min.key </pre> <div class="mw-heading mw-heading4"><h4 id="Maximum">Maximum</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=12" title="Edit section: Maximum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <pre>findMaximum(node) <b>if</b> node is <i>NULL</i> <b>return</b> <i>EMPTY_TREE</i> max := node <b>while</b> max.right is not <i>NULL</i> max := max.right <b>return</b> max.key </pre> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Trie" title="Trie">Trie</a></li> <li><a href="/wiki/Binary_tree" title="Binary tree">Binary tree</a></li> <li><a href="/wiki/Depth-first_search" title="Depth-first search">Depth-first search</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Search_tree&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Black, Paul and Pieterse, Vreda (2005). <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/searchtree.html">"search tree"</a>. <a rel="nofollow" class="external text" href="http://xlinux.nist.gov/dads//">Dictionary of Algorithms and Data Structures</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Toal, Ray. <a rel="nofollow" class="external text" href="http://cs.lmu.edu/~ray/notes/abtrees/">"(a,b) Trees"</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Gildea, Dan (2004). <a rel="nofollow" class="external text" href="http://www.cs.rochester.edu/~gildea/csc282/slides/C12-bst.pdf">"Binary Search Tree"</a></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output 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.navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:CS_trees" title="Template:CS trees"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:CS_trees" title="Template talk:CS trees"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:CS_trees" title="Special:EditPage/Template:CS trees"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Tree_data_structures237" style="font-size:114%;margin:0 4em"><a href="/wiki/Tree_(abstract_data_type)" title="Tree (abstract data type)">Tree data structures</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Search trees</a><br />(<a href="/wiki/Set_(abstract_data_type)" title="Set (abstract data type)">dynamic sets</a>/<a href="/wiki/Associative_array" title="Associative array">associative arrays</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/2%E2%80%933_tree" title="2–3 tree">2–3</a></li> <li><a href="/wiki/2%E2%80%933%E2%80%934_tree" title="2–3–4 tree">2–3–4</a></li> <li><a href="/wiki/AA_tree" title="AA tree">AA</a></li> <li><a href="/wiki/(a,b)-tree" class="mw-redirect" title="(a,b)-tree">(a,b)</a></li> <li><a href="/wiki/AVL_tree" title="AVL tree">AVL</a></li> <li><a href="/wiki/B-tree" title="B-tree">B</a></li> <li><a href="/wiki/B%2B_tree" title="B+ tree">B+</a></li> <li><a href="/wiki/B*-tree" class="mw-redirect" title="B*-tree">B*</a></li> <li><a href="/wiki/Bx-tree" title="Bx-tree">B<sup>x</sup></a></li> <li>(<a href="/wiki/Optimal_binary_search_tree" title="Optimal binary search tree">Optimal</a>) <a href="/wiki/Binary_search_tree" title="Binary search tree">Binary search</a></li> <li><a href="/wiki/Dancing_tree" title="Dancing tree">Dancing</a></li> <li><a href="/wiki/HTree" title="HTree">HTree</a></li> <li><a href="/wiki/Interval_tree" title="Interval tree">Interval</a></li> <li><a href="/wiki/Order_statistic_tree" title="Order statistic tree">Order statistic</a></li> <li><a href="/wiki/Palindrome_tree" title="Palindrome tree">Palindrome</a></li> <li>(<a href="/wiki/Left-leaning_red%E2%80%93black_tree" title="Left-leaning red–black tree">Left-leaning</a>) <a href="/wiki/Red%E2%80%93black_tree" title="Red–black tree">Red–black</a></li> <li><a href="/wiki/Scapegoat_tree" title="Scapegoat tree">Scapegoat</a></li> <li><a href="/wiki/Splay_tree" title="Splay tree">Splay</a></li> <li><a href="/wiki/T-tree" title="T-tree">T</a></li> <li><a href="/wiki/Treap" title="Treap">Treap</a></li> <li><a href="/wiki/UB-tree" title="UB-tree">UB</a></li> <li><a href="/wiki/Weight-balanced_tree" title="Weight-balanced tree">Weight-balanced</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Heap_(data_structure)" title="Heap (data structure)">Heaps</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_heap" title="Binary heap">Binary</a></li> <li><a href="/wiki/Binomial_heap" title="Binomial heap">Binomial</a></li> <li><a href="/wiki/Brodal_queue" title="Brodal queue">Brodal</a></li> <li><a href="/wiki/D-ary_heap" title="D-ary heap"><i>d</i>-ary</a></li> <li><a href="/wiki/Fibonacci_heap" title="Fibonacci heap">Fibonacci</a></li> <li><a href="/wiki/Leftist_tree" title="Leftist tree">Leftist</a></li> <li><a href="/wiki/Pairing_heap" title="Pairing heap">Pairing</a></li> <li><a href="/wiki/Skew_binomial_heap" title="Skew binomial heap">Skew binomial</a></li> <li><a href="/wiki/Skew_heap" title="Skew heap">Skew</a></li> <li><a href="/wiki/Van_Emde_Boas_tree" title="Van Emde Boas tree">van Emde Boas</a></li> <li><a href="/wiki/Weak_heap" title="Weak heap">Weak</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Trie" title="Trie">Tries</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ctrie" title="Ctrie">Ctrie</a></li> <li><a href="/wiki/C-trie" title="C-trie">C-trie (compressed ADT)</a></li> <li><a href="/wiki/Hash_tree_(persistent_data_structure)" title="Hash tree (persistent data structure)">Hash</a></li> <li><a href="/wiki/Radix_tree" title="Radix tree">Radix</a></li> <li><a href="/wiki/Suffix_tree" title="Suffix tree">Suffix</a></li> <li><a href="/wiki/Ternary_search_tree" title="Ternary search tree">Ternary search</a></li> <li><a href="/wiki/X-fast_trie" title="X-fast trie">X-fast</a></li> <li><a href="/wiki/Y-fast_trie" title="Y-fast trie">Y-fast</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spatial_index" class="mw-redirect" title="Spatial index">Spatial</a> data partitioning trees</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ball_tree" title="Ball tree">Ball</a></li> <li><a href="/wiki/BK-tree" title="BK-tree">BK</a></li> <li><a href="/wiki/BSP_tree" class="mw-redirect" title="BSP tree">BSP</a></li> <li><a href="/wiki/Cartesian_tree" title="Cartesian tree">Cartesian</a></li> <li><a href="/wiki/Hilbert_R-tree" title="Hilbert R-tree">Hilbert R</a></li> <li><a href="/wiki/K-d_tree" title="K-d tree"><i>k</i>-d</a> (<a href="/wiki/Implicit_k-d_tree" title="Implicit k-d tree">implicit <i>k</i>-d</a>)</li> <li><a href="/wiki/M-tree" title="M-tree">M</a></li> <li><a href="/wiki/Metric_tree" title="Metric tree">Metric</a></li> <li><a href="/wiki/MVP_tree" class="mw-redirect" title="MVP tree">MVP</a></li> <li><a href="/wiki/Octree" title="Octree">Octree</a></li> 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title="Fenwick tree">Fenwick</a></li> <li><a href="/wiki/Finger_tree" title="Finger tree">Finger</a></li> <li><a href="/wiki/Fractal_tree_index" title="Fractal tree index">Fractal tree index</a></li> <li><a href="/wiki/Fusion_tree" title="Fusion tree">Fusion</a></li> <li><a href="/wiki/Hash_calendar" title="Hash calendar">Hash calendar</a></li> <li><a href="/wiki/IDistance" title="IDistance">iDistance</a></li> <li><a href="/wiki/K-ary_tree" class="mw-redirect" title="K-ary tree">K-ary</a></li> <li><a href="/wiki/Left-child_right-sibling_binary_tree" title="Left-child right-sibling binary tree">Left-child right-sibling</a></li> <li><a href="/wiki/Link/cut_tree" title="Link/cut tree">Link/cut</a></li> <li><a href="/wiki/Log-structured_merge-tree" title="Log-structured merge-tree">Log-structured merge</a></li> <li><a href="/wiki/Merkle_tree" title="Merkle tree">Merkle</a></li> <li><a href="/wiki/PQ_tree" title="PQ tree">PQ</a></li> <li><a href="/wiki/Range_tree" title="Range tree">Range</a></li> <li><a href="/wiki/SPQR_tree" title="SPQR tree">SPQR</a></li> <li><a href="/wiki/Top_tree" title="Top tree">Top</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox authority-control" aria-label="Navbox390" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q621673#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" 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