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[r]" accesskey="r"><span>Ostatnie zmiany</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Wikipedia:Strona_g%C5%82%C3%B3wna" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="wolna encyklopedia" src="/static/images/mobile/copyright/wikipedia-tagline-pl.svg" width="120" height="13" style="width: 7.5em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Specjalna:Szukaj" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Przeszukaj Wikipedię [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Szukaj</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Przeszukaj Wikipedię" aria-label="Przeszukaj Wikipedię" autocapitalize="sentences" title="Przeszukaj Wikipedię [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Specjalna:Szukaj"> </div> <button class="cdx-button cdx-search-input__end-button">Szukaj</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Narzędzia osobiste"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Wygląd"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Zmień rozmiar czcionki, szerokość oraz kolorystykę strony" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Wygląd" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Wygląd</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_pl.wikipedia.org&uselang=pl" class=""><span>Wspomóż Wikipedię</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Specjalna:Utw%C3%B3rz_konto&returnto=Drzewo+binarne" title="Zachęcamy do stworzenia konta i zalogowania, ale nie jest to obowiązkowe." class=""><span>Utwórz konto</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Specjalna:Zaloguj&returnto=Drzewo+binarne" title="Zachęcamy do zalogowania się, choć nie jest to obowiązkowe. 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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Zaloguj się</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Strony dla anonimowych edytorów <a href="/wiki/Pomoc:Pierwsze_kroki" aria-label="Dowiedz się więcej na temat edytowania"><span>dowiedz się więcej</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Specjalna:M%C3%B3j_wk%C5%82ad" title="Lista edycji wykonanych z tego adresu IP [y]" accesskey="y"><span>Edycje</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Specjalna:Moja_dyskusja" title="Dyskusja użytkownika dla tego adresu IP [n]" accesskey="n"><span>Dyskusja</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Witryna"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Spis treści" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Spis treści</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ukryj</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">Początek</div> </a> </li> <li id="toc-Własności" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Własności"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Własności</span> </div> </a> <ul id="toc-Własności-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zobacz_też" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zobacz_też"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Zobacz też</span> </div> </a> <ul id="toc-Zobacz_też-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="Spis treści" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Przełącz stan spisu treści" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Przełącz stan spisu treści</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Drzewo binarne</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Przejdź do artykułu w innym języku. Treść dostępna w 40 językach" > <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-40" 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">40 języków</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B4%D8%AC%D8%B1%D8%A9_%D8%AB%D9%86%D8%A7%D8%A6%D9%8A%D8%A9" title="شجرة ثنائية – arabski" lang="ar" hreflang="ar" data-title="شجرة ثنائية" data-language-autonym="العربية" data-language-local-name="arabski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0kilik_a%C4%9Fac" title="İkilik ağac – azerbejdżański" lang="az" hreflang="az" data-title="İkilik ağac" data-language-autonym="Azərbaycanca" data-language-local-name="azerbejdżański" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B2%D0%BE%D0%B8%D1%87%D0%BD%D0%BE_%D0%B4%D1%8A%D1%80%D0%B2%D0%BE" title="Двоично дърво – bułgarski" lang="bg" hreflang="bg" data-title="Двоично дърво" data-language-autonym="Български" data-language-local-name="bułgarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Binarno_stablo" title="Binarno stablo – bośniacki" lang="bs" hreflang="bs" data-title="Binarno stablo" data-language-autonym="Bosanski" data-language-local-name="bośniacki" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Arbre_binari" title="Arbre binari – kataloński" lang="ca" hreflang="ca" data-title="Arbre binari" data-language-autonym="Català" data-language-local-name="kataloński" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bin%C3%A1rn%C3%AD_strom" title="Binární strom – czeski" lang="cs" hreflang="cs" data-title="Binární strom" data-language-autonym="Čeština" data-language-local-name="czeski" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bin%C3%A4rbaum" title="Binärbaum – niemiecki" lang="de" hreflang="de" data-title="Binärbaum" data-language-autonym="Deutsch" data-language-local-name="niemiecki" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kahendpuu" title="Kahendpuu – estoński" lang="et" hreflang="et" data-title="Kahendpuu" data-language-autonym="Eesti" data-language-local-name="estoński" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CF%85%CE%B1%CE%B4%CE%B9%CE%BA%CF%8C_%CE%B4%CE%AD%CE%BD%CF%84%CF%81%CE%BF" title="Δυαδικό δέντρο – grecki" lang="el" hreflang="el" data-title="Δυαδικό δέντρο" data-language-autonym="Ελληνικά" data-language-local-name="grecki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Binary_tree" title="Binary tree – angielski" lang="en" hreflang="en" data-title="Binary tree" data-language-autonym="English" data-language-local-name="angielski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81rbol_binario" title="Árbol binario – hiszpański" lang="es" hreflang="es" data-title="Árbol binario" data-language-autonym="Español" data-language-local-name="hiszpański" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Duuma_arbo" title="Duuma arbo – esperanto" lang="eo" hreflang="eo" data-title="Duuma arbo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</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%AF%D9%88%D8%AF%D9%88%DB%8C%DB%8C" title="درخت دودویی – perski" lang="fa" hreflang="fa" data-title="درخت دودویی" data-language-autonym="فارسی" data-language-local-name="perski" 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_binaire" title="Arbre binaire – francuski" lang="fr" hreflang="fr" data-title="Arbre binaire" data-language-autonym="Français" data-language-local-name="francuski" 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/%EC%9D%B4%EC%A7%84_%ED%8A%B8%EB%A6%AC" title="이진 트리 – koreański" lang="ko" hreflang="ko" data-title="이진 트리" data-language-autonym="한국어" data-language-local-name="koreański" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A4%BE%E0%A4%87%E0%A4%A8%E0%A4%B0%E0%A5%80_%E0%A4%9F%E0%A5%8D%E0%A4%B0%E0%A5%80" title="बाइनरी ट्री – hindi" lang="hi" hreflang="hi" data-title="बाइनरी ट्री" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Binarno_stablo" title="Binarno stablo – chorwacki" lang="hr" hreflang="hr" data-title="Binarno stablo" data-language-autonym="Hrvatski" data-language-local-name="chorwacki" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pohon_biner" title="Pohon biner – indonezyjski" lang="id" hreflang="id" data-title="Pohon biner" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezyjski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tv%C3%ADundatr%C3%A9" title="Tvíundatré – islandzki" lang="is" hreflang="is" data-title="Tvíundatré" data-language-autonym="Íslenska" data-language-local-name="islandzki" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Albero_binario" title="Albero binario – włoski" lang="it" hreflang="it" data-title="Albero binario" data-language-autonym="Italiano" data-language-local-name="włoski" 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%91%D7%99%D7%A0%D7%90%D7%A8%D7%99" title="עץ בינארי – hebrajski" lang="he" hreflang="he" data-title="עץ בינארי" data-language-autonym="עברית" data-language-local-name="hebrajski" 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" title="Alber binari – lombardzki" lang="lmo" hreflang="lmo" data-title="Alber binari" data-language-autonym="Lombard" data-language-local-name="lombardzki" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D0%BE%D1%91%D1%80%D1%82%D1%8B%D0%BD_%D0%BC%D0%BE%D0%B4" title="Хоёртын мод – mongolski" lang="mn" hreflang="mn" data-title="Хоёртын мод" data-language-autonym="Монгол" data-language-local-name="mongolski" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8C%E5%88%86%E6%9C%A8" title="二分木 – japoński" lang="ja" hreflang="ja" data-title="二分木" data-language-autonym="日本語" data-language-local-name="japoński" 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" title="Árvore binária – portugalski" lang="pt" hreflang="pt" data-title="Árvore binária" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Arbore_binar" title="Arbore binar – rumuński" lang="ro" hreflang="ro" data-title="Arbore binar" data-language-autonym="Română" data-language-local-name="rumuński" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B2%D0%BE%D0%B8%D1%87%D0%BD%D0%BE%D0%B5_%D0%B4%D0%B5%D1%80%D0%B5%D0%B2%D0%BE" title="Двоичное дерево – rosyjski" lang="ru" hreflang="ru" data-title="Двоичное дерево" data-language-autonym="Русский" data-language-local-name="rosyjski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Pem%C3%ABt_binare" title="Pemët binare – albański" lang="sq" hreflang="sq" data-title="Pemët binare" data-language-autonym="Shqip" data-language-local-name="albański" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Binary_tree" title="Binary tree – Simple English" lang="en-simple" hreflang="en-simple" data-title="Binary tree" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Bin%C3%A1rny_strom_(te%C3%B3ria_grafov)" title="Binárny strom (teória grafov) – słowacki" lang="sk" hreflang="sk" data-title="Binárny strom (teória grafov)" data-language-autonym="Slovenčina" data-language-local-name="słowacki" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Dvoji%C5%A1ko_drevo" title="Dvojiško drevo – słoweński" lang="sl" hreflang="sl" data-title="Dvojiško drevo" data-language-autonym="Slovenščina" data-language-local-name="słoweński" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%BE_%D1%81%D1%82%D0%B0%D0%B1%D0%BB%D0%BE" title="Бинарно стабло – serbski" lang="sr" hreflang="sr" data-title="Бинарно стабло" data-language-autonym="Српски / srpski" data-language-local-name="serbski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Binarno_stablo" title="Binarno stablo – serbsko-chorwacki" lang="sh" hreflang="sh" data-title="Binarno stablo" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbsko-chorwacki" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bin%C3%A4%C3%A4ripuu" title="Binääripuu – fiński" lang="fi" hreflang="fi" data-title="Binääripuu" data-language-autonym="Suomi" data-language-local-name="fiński" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bin%C3%A4rtr%C3%A4d" title="Binärträd – szwedzki" lang="sv" hreflang="sv" data-title="Binärträd" data-language-autonym="Svenska" data-language-local-name="szwedzki" class="interlanguage-link-target"><span>Svenska</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%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" title="ต้นไม้แบบทวิภาค – tajski" lang="th" hreflang="th" data-title="ต้นไม้แบบทวิภาค" data-language-autonym="ไทย" data-language-local-name="tajski" 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%94%D0%B2%D1%96%D0%B9%D0%BA%D0%BE%D0%B2%D0%B5_%D0%B4%D0%B5%D1%80%D0%B5%D0%B2%D0%BE" title="Двійкове дерево – ukraiński" lang="uk" hreflang="uk" data-title="Двійкове дерево" data-language-autonym="Українська" data-language-local-name="ukraiński" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C3%A2y_nh%E1%BB%8B_ph%C3%A2n" title="Cây nhị phân – 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accesskey="g"><span>Element Wikidanych</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Narzędzia dla stron"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Wygląd"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Wygląd</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ukryj</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" 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">Z Wikipedii, wolnej encyklopedii</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="pl" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Plik:Binary_tree.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Binary_tree.svg/220px-Binary_tree.svg.png" decoding="async" width="220" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Binary_tree.svg/330px-Binary_tree.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Binary_tree.svg/440px-Binary_tree.svg.png 2x" data-file-width="300" data-file-height="250" /></a><figcaption>Przykładowe drzewo binarne o rozmiarze 9 i wysokości 3</figcaption></figure> <p><b>Drzewo binarne</b> – <a href="/wiki/Drzewo_(matematyka)" title="Drzewo (matematyka)">drzewo</a>, w którym <a href="/wiki/Stopie%C5%84_wierzcho%C5%82ka" title="Stopień wierzchołka">stopień</a> każdego wierzchołka jest nie większy od 3. </p><p><b>Ukorzenione drzewo binarne</b> to drzewo binarne, w którym wyróżniono jeden z wierzchołków (zwany <i>korzeniem</i>) stopnia najwyżej 2. </p><p>W <a href="/wiki/Informatyka" title="Informatyka">informatyce</a> drzewo binarne to jeden z rodzajów <a href="/wiki/Drzewo_(informatyka)" title="Drzewo (informatyka)">drzewa</a> (<a href="/wiki/Struktura_danych" title="Struktura danych">struktury danych</a>), w którym liczba synów każdego wierzchołka wynosi <b>nie więcej</b> niż dwa. Wyróżnia się wtedy <b>lewego syna</b> i <b>prawego syna</b> danego wierzchołka. </p><p>Drzewo binarne, w którym liczba synów każdego wierzchołka wynosi albo zero albo dwa, nazywane jest <b>drzewem regularnym</b>. Przykładem takich drzew są <a href="/wiki/Kodowanie_Huffmana" title="Kodowanie Huffmana">drzewa Huffmana</a>. </p><p>Szczególnymi odmianami drzew binarnych są <a href="/wiki/Binarne_drzewo_poszukiwa%C5%84" title="Binarne drzewo poszukiwań">drzewa BST</a>, <a href="/wiki/Drzewo_BSP" class="mw-redirect" title="Drzewo BSP">drzewa BSP</a> oraz <a href="/wiki/Kopiec_binarny" title="Kopiec binarny">kopce binarne</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Własności"><span id="W.C5.82asno.C5.9Bci"></span>Własności</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Drzewo_binarne&veaction=edit&section=1" title="Edytuj sekcję: Własności" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Drzewo_binarne&action=edit&section=1" title="Edytuj kod źródłowy sekcji: Własności"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Liczba n-wierzchołkowych ukorzenionych drzew binarnych wynosi: </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 b_{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e891f2e26ca623a2f9bbd0bfef87a4dfac0a20f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.313ex; height:2.509ex;" alt="{\displaystyle b_{0}=1}"></span></dd> <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 b_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50df2603dcbbd4164de6c801e761c25e93d6b810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.313ex; height:2.509ex;" alt="{\displaystyle b_{1}=1}"></span></dd> <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 b_{n}=\sum _{j=0}^{n-1}b_{j}b_{n-1-j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}=\sum _{j=0}^{n-1}b_{j}b_{n-1-j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997bd9b2cfcb01e3d342bb50d57ae546283a7ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.236ex; height:7.676ex;" alt="{\displaystyle b_{n}=\sum _{j=0}^{n-1}b_{j}b_{n-1-j}}"></span></dd></dl> <p>Istnieje też postać zwarta: </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 b_{n}={\frac {1}{n+1}}{2n \choose n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}={\frac {1}{n+1}}{2n \choose n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a40b79cf41152b0e1fb488b4736ebd31980bd2e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.527ex; height:6.176ex;" alt="{\displaystyle b_{n}={\frac {1}{n+1}}{2n \choose n}}"></span></dd></dl> <p>znana jako <a href="/w/index.php?title=Rekursywna_relacja_Catalana&action=edit&redlink=1" class="new" title="Rekursywna relacja Catalana (strona nie istnieje)">rekursywna relacja Catalana</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Zobacz_też"><span id="Zobacz_te.C5.BC"></span>Zobacz też</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Drzewo_binarne&veaction=edit&section=2" title="Edytuj sekcję: Zobacz też" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Drzewo_binarne&action=edit&section=2" title="Edytuj kod źródłowy sekcji: Zobacz też"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="infobox noprint plainlinks"> <tbody><tr> <td style="text-align: center; font-weight: bold;">Informacje w <a href="/wiki/Wikipedia:Projekty_siostrzane" title="Wikipedia:Projekty siostrzane">projektach siostrzanych</a></td> </tr> <tr> <td><span style="margin:1px"><span typeof="mw:File"><a href="/wiki/Plik:Commons-logo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/13px-Commons-logo.svg.png" decoding="async" width="13" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/26px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></span>  <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Binary_trees?uselang=pl"><b>Multimedia</b></a></span> w <a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commons</a></td> </tr><tr> <td><span typeof="mw:File"><a href="/wiki/Plik:WiktionaryPl_nodesc.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/15px-WiktionaryPl_nodesc.svg.png" decoding="async" width="15" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/23px-WiktionaryPl_nodesc.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/30px-WiktionaryPl_nodesc.svg.png 2x" data-file-width="122" data-file-height="117" /></a></span>  <b><a href="https://pl.wiktionary.org/wiki/drzewo_binarne" class="extiw" title="wikt:drzewo binarne">Definicje słownikowe</a></b> w <a href="/wiki/Wikis%C5%82ownik" title="Wikisłownik">Wikisłowniku</a></td> </tr> </tbody></table> <ul><li><a href="/wiki/Przechodzenie_drzewa" title="Przechodzenie drzewa">przechodzenie drzewa</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r74016753">.mw-parser-output #normdaten>div+div{margin-top:0.5em}.mw-parser-output #normdaten>div>div{background:var(--background-color-neutral,#eaecf0);padding:.2em .5em}.mw-parser-output #normdaten ul{margin:0;padding:0}.mw-parser-output #normdaten ul li:first-child{padding-left:.5em;border-left:1px solid var(--border-color-base,#a2a9b1)}</style> <div id="normdaten" class="catlinks"><div class="normdaten-typ-fehlt"><div><a href="/wiki/Kontrola_autorytatywna" title="Kontrola autorytatywna">Kontrola autorytatywna</a> (<span class="description"><a href="/wiki/Abstrakcyjny_typ_danych" title="Abstrakcyjny typ danych">abstrakcyjny typ danych</a></span>):</div><ul><li><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://d-nb.info/gnd/4145532-0">4145532-0</a></span></li></ul></div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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