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Available in 13 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-13" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">13 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81rbol_binario_de_b%C3%BAsqueda_auto-balanceable" title="Árbol binario de búsqueda auto-balanceable – Spanish" lang="es" hreflang="es" data-title="Árbol binario de búsqueda auto-balanceable" 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%AF%D8%B1%D8%AE%D8%AA_%D8%AC%D8%B3%D8%AA%D8%AC%D9%88%DB%8C_%D8%AF%D9%88%D8%AF%D9%88%DB%8C%DB%8C_%D8%AE%D9%88%D8%AF-%D9%85%D8%AA%D9%88%D8%A7%D8%B2%D9%86" title="درخت جستجوی دودویی خود-متوازن – Persian" lang="fa" hreflang="fa" data-title="درخت جستجوی دودویی خود-متوازن" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EA%B0%80_%EA%B7%A0%ED%98%95_%EC%9D%B4%EC%A7%84_%ED%83%90%EC%83%89_%ED%8A%B8%EB%A6%AC" title="자가 균형 이진 탐색 트리 – Korean" lang="ko" hreflang="ko" data-title="자가 균형 이진 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Albero_binario_di_ricerca_bilanciato" title="Albero binario di ricerca bilanciato – Italian" lang="it" hreflang="it" data-title="Albero binario di ricerca bilanciato" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A5_%D7%9E%D7%90%D7%95%D7%96%D7%9F" title="עץ מאוזן – Hebrew" lang="he" hreflang="he" data-title="עץ מאוזן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Alber_binari_de_ricerca_bilanciaa" title="Alber binari de ricerca bilanciaa – Lombard" lang="lmo" hreflang="lmo" data-title="Alber binari de ricerca bilanciaa" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%B3%E8%A1%A1%E4%BA%8C%E5%88%86%E6%8E%A2%E7%B4%A2%E6%9C%A8" title="平衡二分探索木 – Japanese" lang="ja" hreflang="ja" data-title="平衡二分探索木" 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_bin%C3%A1ria_de_busca_balanceada" title="Árvore binária de busca balanceada – Portuguese" lang="pt" hreflang="pt" data-title="Árvore binária de busca balanceada" 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/%D0%A1%D0%B0%D0%BC%D0%BE%D0%B1%D0%B0%D0%BB%D0%B0%D0%BD%D1%81%D0%B8%D1%80%D0%B0%D1%98%D1%83%D1%9B%D0%B5_%D0%B1%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%BE_%D1%81%D1%82%D0%B0%D0%B1%D0%BB%D0%BE_%D0%BF%D1%80%D0%B5%D1%82%D1%80%D0%B0%D0%B3%D0%B5" title="Самобалансирајуће бинарно стабло претраге – Serbian" lang="sr" hreflang="sr" data-title="Самобалансирајуће бинарно стабло претраге" 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-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%95%E0%B9%89%E0%B8%99%E0%B9%84%E0%B8%A1%E0%B9%89%E0%B8%84%E0%B9%89%E0%B8%99%E0%B8%AB%E0%B8%B2%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%97%E0%B8%A7%E0%B8%B4%E0%B8%A0%E0%B8%B2%E0%B8%84%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B8%A1%E0%B8%B5%E0%B9%82%E0%B8%84%E0%B8%A3%E0%B8%87%E0%B8%AA%E0%B8%A3%E0%B9%89%E0%B8%B2%E0%B8%87%E0%B8%9B%E0%B8%A3%E0%B8%B1%E0%B8%9A%E0%B8%AA%E0%B8%A1%E0%B8%94%E0%B8%B8%E0%B8%A5%E0%B9%80%E0%B8%AD%E0%B8%87%E0%B9%84%E0%B8%94%E0%B9%89" title="ต้นไม้ค้นหาแบบทวิภาคที่มีโครงสร้างปรับสมดุลเองได้ – Thai" lang="th" hreflang="th" data-title="ต้นไม้ค้นหาแบบทวิภาคที่มีโครงสร้างปรับสมดุลเองได้" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B1%D0%B0%D0%BB%D0%B0%D0%BD%D1%81%D0%BE%D0%B2%D0%B0%D0%BD%D0%B5_%D0%B4%D0%B5%D1%80%D0%B5%D0%B2%D0%BE" title="Збалансоване дерево – Ukrainian" lang="uk" hreflang="uk" data-title="Збалансоване дерево" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%87%AA%E5%B9%B3%E8%A1%A1%E4%BA%8C%E5%85%83%E6%90%9C%E5%B0%8B%E6%A8%B9" title="自平衡二元搜尋樹 – Cantonese" lang="yue" hreflang="yue" data-title="自平衡二元搜尋樹" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B9%B3%E8%A1%A1%E6%A0%91" title="平衡树 – Chinese" lang="zh" hreflang="zh" data-title="平衡树" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Any node-based binary search tree that automatically keeps its height the same</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Self-balancing+binary+search+tree%22">"Self-balancing binary search tree"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Self-balancing+binary+search+tree%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Self-balancing+binary+search+tree%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Self-balancing+binary+search+tree%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Self-balancing+binary+search+tree%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Self-balancing+binary+search+tree%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">November 2010</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Unbalanced_binary_tree.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Unbalanced_binary_tree.svg/251px-Unbalanced_binary_tree.svg.png" decoding="async" width="251" height="251" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Unbalanced_binary_tree.svg/377px-Unbalanced_binary_tree.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Unbalanced_binary_tree.svg/502px-Unbalanced_binary_tree.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>An example of an <b>unbalanced</b> tree; following the path from the root to a node takes an average of 3.27 node accesses</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:AVLtreef.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/AVLtreef.svg/251px-AVLtreef.svg.png" decoding="async" width="251" height="112" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/AVLtreef.svg/377px-AVLtreef.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/AVLtreef.svg/502px-AVLtreef.svg.png 2x" data-file-width="2223" data-file-height="990" /></a><figcaption>The same tree after being height-balanced; the average path effort decreased to 3.00 node accesses</figcaption></figure> <p>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, a <b>self-balancing binary search tree</b> (BST) is any <a href="/wiki/Node_(computer_science)" title="Node (computer science)">node</a>-based <a href="/wiki/Binary_search_tree" title="Binary search tree">binary search tree</a> that automatically keeps its height (maximal number of levels below the root) small in the face of arbitrary item insertions and deletions.<sup id="cite_ref-knuth_1-0" class="reference"><a href="#cite_note-knuth-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> These operations when designed for a self-balancing binary search tree, contain precautionary measures against boundlessly increasing tree height, so that these <a href="/wiki/Abstract_data_type" title="Abstract data type">abstract data structures</a> receive the attribute "self-balancing". </p><p>For <b>height-balanced</b> binary trees, the height is defined to be logarithmic <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 O(\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae0f22048ba6b7c05dbae17b056bfa16e21807d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle O(\log n)}"></span> in the number <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> of items. This is the case for many binary search trees, such as <a href="/wiki/AVL_tree" title="AVL tree">AVL trees</a> and <a href="/wiki/Red%E2%80%93black_tree" title="Red–black tree">red–black trees</a>. <a href="/wiki/Splay_tree" title="Splay tree">Splay trees</a> and <a href="/wiki/Treap" title="Treap">treaps</a> are self-balancing but not height-balanced, as their height is not guaranteed to be logarithmic in the number of items. </p><p>Self-balancing binary search trees provide efficient implementations for mutable ordered <a href="/wiki/List_(computing)" class="mw-redirect" title="List (computing)">lists</a>, and can be used for other abstract data structures such as <a href="/wiki/Associative_array" title="Associative array">associative arrays</a>, <a href="/wiki/Priority_queue" title="Priority queue">priority queues</a> and <a href="/wiki/Set_(abstract_data_type)" title="Set (abstract data type)">sets</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-balancing_binary_search_tree&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:BinaryTreeRotations.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/BinaryTreeRotations.svg/300px-BinaryTreeRotations.svg.png" decoding="async" width="300" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/BinaryTreeRotations.svg/450px-BinaryTreeRotations.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/BinaryTreeRotations.svg/600px-BinaryTreeRotations.svg.png 2x" data-file-width="405" data-file-height="224" /></a><figcaption>Tree rotations are very common internal operations on self-balancing binary trees to keep perfect or near-to-perfect balance.</figcaption></figure> <p>Most operations on a binary search tree (BST) take time directly proportional to the height of the tree, so it is desirable to keep the height small. A binary tree with height <i>h</i> can contain at most <a href="/wiki/Geometric_series#Closed-form_formula" title="Geometric series">2<sup>0</sup>+2<sup>1</sup>+···+2<sup><i>h</i></sup> = 2<sup><i>h</i>+1</sup>−1</a> nodes. It follows that for any tree with <i>n</i> nodes and height <i>h</i>: </p> <dl><dd><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 n\leq 2^{h+1}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≤</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\leq 2^{h+1}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34bfb3725cd0fd6e834537810c74080e0e2180d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.938ex; height:2.843ex;" alt="{\displaystyle n\leq 2^{h+1}-1}"></span></dd></dl> <p>And that implies: </p> <dl><dd><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 h\geq \lceil \log _{2}(n+1)-1\rceil \geq \lfloor \log _{2}n\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>≥</mo> <mo fence="false" stretchy="false">⌈</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌉</mo> <mo>≥</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\geq \lceil \log _{2}(n+1)-1\rceil \geq \lfloor \log _{2}n\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4264fcbd5d567771faa2bd486de80e2593a07575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.709ex; height:2.843ex;" alt="{\displaystyle h\geq \lceil \log _{2}(n+1)-1\rceil \geq \lfloor \log _{2}n\rfloor }"></span>.</dd></dl> <p>In other words, the minimum height of a binary tree with <i>n</i> nodes is <span class="nowrap"><a href="/wiki/Logarithm" title="Logarithm">log</a><sub>2</sub>(<i>n</i>),</span> <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">rounded down</a>; that is, <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 \lfloor \log _{2}n\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor \log _{2}n\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992c9e2c99d88241b484aad58ab2f6875708d2fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.873ex; height:2.843ex;" alt="{\displaystyle \lfloor \log _{2}n\rfloor }"></span>.<sup id="cite_ref-knuth_1-1" class="reference"><a href="#cite_note-knuth-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>However, the simplest algorithms for BST item insertion may yield a tree with height <i>n</i> in rather common situations. For example, when the items are inserted in sorted <a href="/wiki/Key_(database)" class="mw-redirect" title="Key (database)">key</a> order, the tree degenerates into a <a href="/wiki/Linked_list" title="Linked list">linked list</a> with <i>n</i> nodes. The difference in performance between the two situations may be enormous: for example, when <i>n</i> = 1,000,000, the minimum height is <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 \lfloor \log _{2}(1,000,000)\rfloor =19}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>000</mn> <mo>,</mo> <mn>000</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>=</mo> <mn>19</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor \log _{2}(1,000,000)\rfloor =19}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73c74b7a3617eaae266b0e5af80ac0bedbf44afc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.529ex; height:2.843ex;" alt="{\displaystyle \lfloor \log _{2}(1,000,000)\rfloor =19}"></span>. </p><p>If the data items are known ahead of time, the height can be kept small, in the average sense, by adding values in a random order, resulting in a <a href="/wiki/Random_binary_search_tree" class="mw-redirect" title="Random binary search tree">random binary search tree</a>. However, there are many situations (such as <a href="/wiki/Online_algorithm" title="Online algorithm">online algorithms</a>) where this <a href="/wiki/Randomized_algorithm" title="Randomized algorithm">randomization</a> is not viable. </p><p>Self-balancing binary trees solve this problem by performing transformations on the tree (such as <a href="/wiki/Tree_rotation" title="Tree rotation">tree rotations</a>) at key insertion times, in order to keep the height proportional to <span class="nowrap">log<sub>2</sub>(<i>n</i>).</span> Although a certain <a href="/wiki/Computational_overhead" class="mw-redirect" title="Computational overhead">overhead</a> is involved, it is not bigger than the always necessary lookup cost and may be justified by ensuring fast execution of all operations. </p><p>While it is possible to maintain a BST with minimum height with expected <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 O(\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae0f22048ba6b7c05dbae17b056bfa16e21807d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle O(\log n)}"></span> time operations (lookup/insertion/removal), the additional space requirements required to maintain such a structure tend to outweigh the decrease in search time. For comparison, an <a href="/wiki/AVL_tree" title="AVL tree">AVL tree</a> is guaranteed to be within a factor of 1.44 of the optimal height while requiring only two additional bits of storage in a naive implementation.<sup id="cite_ref-knuth_1-2" class="reference"><a href="#cite_note-knuth-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Therefore, most self-balancing BST algorithms keep the height within a constant factor of this lower bound. </p><p>In the <a href="/wiki/Asymptotic" class="mw-redirect" title="Asymptotic">asymptotic</a> ("<a href="/wiki/Big_O_notation" title="Big O notation">Big-O</a>") sense, a self-balancing BST structure containing <i>n</i> items allows the lookup, insertion, and removal of an item in <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 O(\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae0f22048ba6b7c05dbae17b056bfa16e21807d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle O(\log n)}"></span> worst-case time, and <a href="/wiki/In-order_iteration" class="mw-redirect" title="In-order iteration">ordered enumeration</a> of all items in <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 O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}"></span> time. For some implementations these are per-operation time bounds, while for others they are <a href="/wiki/Amortized_analysis" title="Amortized analysis">amortized</a> bounds over a sequence of operations. These times are asymptotically optimal among all data structures that manipulate the key only through comparisons. </p> <div class="mw-heading mw-heading2"><h2 id="Implementations">Implementations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-balancing_binary_search_tree&action=edit&section=2" title="Edit section: Implementations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Data structures implementing this type of tree include: </p> <ul><li><a href="/wiki/2%E2%80%933_tree" title="2–3 tree">2–3 tree</a></li> <li><a href="/wiki/AA_tree" title="AA tree">AA tree</a></li> <li><a href="/wiki/AVL_tree" title="AVL tree">AVL tree</a></li> <li><a href="/wiki/B-tree" title="B-tree">B-tree</a></li> <li><a href="/wiki/Red%E2%80%93black_tree" title="Red–black tree">Red–black tree</a></li> <li><a href="/wiki/Scapegoat_tree" title="Scapegoat tree">Scapegoat tree</a></li> <li><a href="/wiki/Tango_tree" title="Tango tree">Tango tree</a></li> <li><a href="/wiki/Treap" title="Treap">Treap</a></li> <li><a href="/wiki/Weight-balanced_tree" title="Weight-balanced tree">Weight-balanced tree</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-balancing_binary_search_tree&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Self-balancing binary search trees can be used in a natural way to construct and maintain ordered lists, such as <a href="/wiki/Priority_queue" title="Priority queue">priority queues</a>. They can also be used for <a href="/wiki/Associative_array" title="Associative array">associative arrays</a>; key-value pairs are simply inserted with an ordering based on the key alone. In this capacity, self-balancing BSTs have <a href="/wiki/Associative_array#Efficient_representations" title="Associative array">a number of advantages and disadvantages</a> over their main competitor, <a href="/wiki/Hash_table" title="Hash table">hash tables</a>. One advantage of self-balancing BSTs is that they allow fast (indeed, asymptotically optimal) enumeration of the items <i>in key order</i>, which hash tables do not provide. One disadvantage is that their lookup algorithms get more complicated when there may be multiple items with the same key. Self-balancing BSTs have better worst-case lookup performance than most<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> hash tables (<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 O(\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae0f22048ba6b7c05dbae17b056bfa16e21807d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle O(\log n)}"></span> compared to <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 O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}"></span>), but have worse average-case performance (<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 O(\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae0f22048ba6b7c05dbae17b056bfa16e21807d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle O(\log n)}"></span> compared to <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 O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span>). </p><p>Self-balancing BSTs can be used to implement any algorithm that requires mutable ordered lists, to achieve optimal worst-case asymptotic performance. For example, if <a href="/wiki/Binary_tree_sort" class="mw-redirect" title="Binary tree sort">binary tree sort</a> is implemented with a self-balancing BST, we have a very simple-to-describe yet <a href="/wiki/Asymptotically_optimal" class="mw-redirect" title="Asymptotically optimal">asymptotically optimal</a> <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 O(n\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2320768fb54880ca4356e61f60eb02a3f9d9f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.118ex; height:2.843ex;" alt="{\displaystyle O(n\log n)}"></span> sorting algorithm. Similarly, many algorithms in <a href="/wiki/Computational_geometry" title="Computational geometry">computational geometry</a> exploit variations on self-balancing BSTs to solve problems such as the <a href="/wiki/Line_segment_intersection" class="mw-redirect" title="Line segment intersection">line segment intersection</a> problem and the <a href="/wiki/Point_location" title="Point location">point location</a> problem efficiently. (For average-case performance, however, self-balancing BSTs may be less efficient than other solutions. Binary tree sort, in particular, is likely to be slower than <a href="/wiki/Merge_sort" title="Merge sort">merge sort</a>, <a href="/wiki/Quicksort" title="Quicksort">quicksort</a>, or <a href="/wiki/Heapsort" title="Heapsort">heapsort</a>, because of the tree-balancing overhead as well as <a href="/wiki/Cache_(computing)" title="Cache (computing)">cache</a> access patterns.) </p><p>Self-balancing BSTs are flexible data structures, in that it's easy to extend them to efficiently record additional information or perform new operations. For example, one can record the number of nodes in each subtree having a certain property, allowing one to count the number of nodes in a certain key range with that property in <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 O(\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae0f22048ba6b7c05dbae17b056bfa16e21807d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle O(\log n)}"></span> time. These extensions can be used, for example, to optimize database queries or other list-processing algorithms. </p> <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=Self-balancing_binary_search_tree&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Search_data_structure" title="Search data structure">Search data structure</a></li> <li><a href="/wiki/Day%E2%80%93Stout%E2%80%93Warren_algorithm" title="Day–Stout–Warren algorithm">Day–Stout–Warren algorithm</a></li> <li><a href="/wiki/Fusion_tree" title="Fusion tree">Fusion tree</a></li> <li><a href="/wiki/Skip_list" title="Skip list">Skip list</a></li> <li><a href="/wiki/Sorting" title="Sorting">Sorting</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=Self-balancing_binary_search_tree&action=edit&section=5" 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-knuth-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-knuth_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-knuth_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-knuth_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a>. <i><a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a></i>, Volume 3: <i>Sorting and Searching</i>, Second Edition. Addison-Wesley, 1998. <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-89685-0" title="Special:BookSources/0-201-89685-0">0-201-89685-0</a>. Section 6.2.3: Balanced Trees, pp.458–481.</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"><a href="/wiki/Cuckoo_hashing" title="Cuckoo hashing">Cuckoo hashing</a> provides worst-case lookup performance of <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 O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Self-balancing_binary_search_tree&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/heightBalancedTree.html">Dictionary of Algorithms and Data Structures: Height-balanced binary search tree</a></li> <li><a rel="nofollow" class="external text" href="http://adtinfo.org/">GNU libavl</a>, a LGPL-licensed library of binary tree implementations in C, with documentation</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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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> <li><a href="/wiki/PH-tree" title="PH-tree">PH</a></li> <li><a href="/wiki/Priority_R-tree" title="Priority R-tree">Priority R</a></li> <li><a href="/wiki/Quadtree" title="Quadtree">Quad</a></li> <li><a href="/wiki/R-tree" title="R-tree">R</a></li> <li><a href="/wiki/R%2B_tree" title="R+ tree">R+</a></li> <li><a href="/wiki/R*_tree" class="mw-redirect" title="R* tree">R*</a></li> <li><a href="/wiki/Segment_tree" title="Segment tree">Segment</a></li> <li><a href="/wiki/Vantage-point_tree" title="Vantage-point tree">VP</a></li> <li><a href="/wiki/X-tree" title="X-tree">X</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other trees</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/Cover_tree" title="Cover tree">Cover</a></li> <li><a href="/wiki/Exponential_tree" title="Exponential tree">Exponential</a></li> <li><a href="/wiki/Fenwick_tree" 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" aria-labelledby="Data_structures" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Data_structures" title="Template:Data structures"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Data_structures" title="Template talk:Data structures"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Data_structures" title="Special:EditPage/Template:Data structures"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Data_structures" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_structure" title="Data structure">Data structures</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Collection_(abstract_data_type)" title="Collection (abstract data type)">Collection</a></li> <li><a href="/wiki/Container_(abstract_data_type)" title="Container (abstract data type)">Container</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Abstract_data_type" title="Abstract data type">Abstract</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Associative_array" title="Associative array">Associative array</a> <ul><li><a href="/wiki/Multimap" title="Multimap">Multimap</a></li> <li><a href="/wiki/Retrieval_Data_Structure" title="Retrieval Data Structure">Retrieval Data Structure</a></li></ul></li> <li><a href="/wiki/List_(abstract_data_type)" title="List (abstract data type)">List</a></li> <li><a href="/wiki/Stack_(abstract_data_type)" title="Stack (abstract data type)">Stack</a></li> <li><a href="/wiki/Queue_(abstract_data_type)" title="Queue (abstract data type)">Queue</a> <ul><li><a href="/wiki/Double-ended_queue" title="Double-ended queue">Double-ended queue</a></li></ul></li> <li><a href="/wiki/Priority_queue" title="Priority queue">Priority queue</a> <ul><li><a href="/wiki/Double-ended_priority_queue" title="Double-ended priority queue">Double-ended priority queue</a></li></ul></li> <li><a href="/wiki/Set_(abstract_data_type)" title="Set (abstract data type)">Set</a> <ul><li><a href="/wiki/Set_(abstract_data_type)#Multiset" title="Set (abstract data type)">Multiset</a></li> <li><a href="/wiki/Disjoint-set_data_structure" title="Disjoint-set data structure">Disjoint-set</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Array_(data_structure)" title="Array (data structure)">Arrays</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit_array" title="Bit array">Bit array</a></li> <li><a href="/wiki/Circular_buffer" title="Circular buffer">Circular buffer</a></li> <li><a href="/wiki/Dynamic_array" title="Dynamic array">Dynamic array</a></li> <li><a href="/wiki/Hash_table" title="Hash table">Hash table</a></li> <li><a href="/wiki/Hashed_array_tree" title="Hashed array tree">Hashed array tree</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse matrix</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linked_data_structure" title="Linked data structure">Linked</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Association_list" title="Association list">Association list</a></li> <li><a href="/wiki/Linked_list" title="Linked list">Linked list</a></li> <li><a href="/wiki/Skip_list" title="Skip list">Skip list</a></li> <li><a href="/wiki/Unrolled_linked_list" title="Unrolled linked list">Unrolled linked list</a></li> <li><a href="/wiki/XOR_linked_list" title="XOR linked list">XOR linked list</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tree_(data_structure)" class="mw-redirect" title="Tree (data structure)">Trees</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/B-tree" title="B-tree">B-tree</a></li> <li><a href="/wiki/Binary_search_tree" title="Binary search tree">Binary search tree</a> <ul><li><a href="/wiki/AA_tree" title="AA tree">AA tree</a></li> <li><a href="/wiki/AVL_tree" title="AVL tree">AVL tree</a></li> <li><a href="/wiki/Red%E2%80%93black_tree" title="Red–black tree">Red–black tree</a></li> <li><a class="mw-selflink selflink">Self-balancing tree</a></li> <li><a href="/wiki/Splay_tree" title="Splay tree">Splay tree</a></li></ul></li> <li><a href="/wiki/Heap_(data_structure)" title="Heap (data structure)">Heap</a> <ul><li><a href="/wiki/Binary_heap" title="Binary heap">Binary heap</a></li> <li><a href="/wiki/Binomial_heap" title="Binomial heap">Binomial heap</a></li> <li><a href="/wiki/Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a></li></ul></li> <li><a href="/wiki/R-tree" title="R-tree">R-tree</a> <ul><li><a href="/wiki/R*_tree" class="mw-redirect" title="R* tree">R* tree</a></li> <li><a href="/wiki/R%2B_tree" title="R+ tree">R+ tree</a></li> <li><a href="/wiki/Hilbert_R-tree" title="Hilbert R-tree">Hilbert R-tree</a></li></ul></li> <li><a href="/wiki/Trie" title="Trie">Trie</a> <ul><li><a href="/wiki/Hash_tree_(persistent_data_structure)" title="Hash tree (persistent data structure)">Hash tree</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Graph_(abstract_data_type)" title="Graph (abstract data type)">Graphs</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_decision_diagram" title="Binary decision diagram">Binary decision diagram</a></li> <li><a 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