A\u011fa\u00e7lar, bilgisayar bilimlerinde ve yaz\u0131l\u0131m m\u00fchendisli\u011finde en temel ve yayg\u0131n kullan\u0131lan do\u011frusal olmayan veri yap\u0131lar\u0131ndan biridir. Hiyerar\u015fik ili\u015fkileri modellemek, verileri d\u00fczenlemek ve h\u0131zl\u0131 eri\u015fim sa\u011flamak i\u00e7in kullan\u0131l\u0131rlar. Bir a\u011fac\u0131n etkinli\u011fi ve performans\u0131, yap\u0131s\u0131n\u0131n belirli \u00f6zelliklerine, \u00f6zellikle de y\u00fcksekli\u011fine ba\u011fl\u0131d\u0131r. Bu makale, a\u011fa\u00e7 veri yap\u0131lar\u0131ndaki y\u00fckseklik kavram\u0131n\u0131 derinlemesine inceleyecek; tan\u0131m\u0131n\u0131, farkl\u0131 hesaplama y\u00f6ntemlerini, \u00e7e\u015fitli a\u011fa\u00e7 t\u00fcrlerindeki y\u00fckseklik karakteristiklerini ve bilgisayar bilimlerindeki kritik \u00f6nemini detayland\u0131racakt\u0131r.<\/p>\n
Veri yap\u0131lar\u0131, verilerin bilgisayar belle\u011finde nas\u0131l d\u00fczenlendi\u011fini ve sakland\u0131\u011f\u0131n\u0131 tan\u0131mlayan temel yap\u0131 ta\u015flar\u0131d\u0131r. A\u011fa\u00e7lar, bu yap\u0131 ta\u015flar\u0131 aras\u0131nda \u00f6zel bir yere sahiptir \u00e7\u00fcnk\u00fc hiyerar\u015fik ili\u015fkileri temsil etme yetenekleri sayesinde dosya sistemlerinden veritaban\u0131 indekslerine, s\u00f6zdizimi ayr\u0131\u015ft\u0131r\u0131c\u0131lar\u0131ndan (syntax parsers) yapay zeka algoritmalar\u0131na kadar geni\u015f bir yelpazede uygulama alan\u0131 bulurlar. Bir a\u011fac\u0131n performans\u0131 ve verimlili\u011fi, genellikle o a\u011fac\u0131n “y\u00fcksekli\u011fi” ile do\u011frudan ili\u015fkilidir. Y\u00fckseklik, bir a\u011fac\u0131n ne kadar “uzun” veya “derin” oldu\u011funu g\u00f6steren bir \u00f6l\u00e7\u00fctt\u00fcr ve arama, ekleme, silme gibi temel i\u015flemlerin zaman karma\u015f\u0131kl\u0131\u011f\u0131n\u0131 belirlemede merkezi bir rol oynar. Bu makalede, a\u011fa\u00e7 y\u00fcksekli\u011finin ne anlama geldi\u011fini, nas\u0131l hesapland\u0131\u011f\u0131n\u0131, farkl\u0131 a\u011fa\u00e7 t\u00fcrlerinde nas\u0131l de\u011fi\u015fti\u011fini ve neden bu kadar \u00f6nemli oldu\u011funu ayr\u0131nt\u0131l\u0131 olarak inceleyece\u011fiz. Amac\u0131m\u0131z, okuyuculara a\u011fa\u00e7 y\u00fcksekli\u011fi hakk\u0131nda kapsaml\u0131 ve teknik bir anlay\u0131\u015f sunmakt\u0131r.<\/p>\n
A\u011fa\u00e7 y\u00fcksekli\u011fi kavram\u0131na ge\u00e7meden \u00f6nce, a\u011fa\u00e7 veri yap\u0131lar\u0131yla ilgili temel terminolojiyi netle\u015ftirmek \u00f6nemlidir. Bu terimler, a\u011fa\u00e7lar\u0131n yap\u0131s\u0131n\u0131 ve bile\u015fenlerini anlamak i\u00e7in anahtard\u0131r:<\/p>\n
Bir a\u011fac\u0131n temel birimidir. Her d\u00fc\u011f\u00fcm bir veri par\u00e7as\u0131n\u0131 saklar ve di\u011fer d\u00fc\u011f\u00fcmlere ba\u011flant\u0131lar i\u00e7erebilir.<\/p>\n
A\u011fa\u00e7taki en \u00fcst d\u00fc\u011f\u00fcmd\u00fcr ve hi\u00e7bir ebeveyni yoktur. Bir a\u011fac\u0131n tek bir k\u00f6k d\u00fc\u011f\u00fcm\u00fc bulunur.<\/p>\n
Bir d\u00fc\u011f\u00fcme do\u011frudan ba\u011fl\u0131 olan alt d\u00fc\u011f\u00fcmlerdir.<\/p>\n
Bir d\u00fc\u011f\u00fcm\u00fcn do\u011frudan \u00fcst\u00fcndeki d\u00fc\u011f\u00fcmd\u00fcr.<\/p>\n
Ayn\u0131 ebeveyn d\u00fc\u011f\u00fcm\u00fcne sahip olan d\u00fc\u011f\u00fcmlerdir.<\/p>\n
Hi\u00e7 \u00e7ocu\u011fu olmayan d\u00fc\u011f\u00fcmlerdir. A\u011fac\u0131n “u\u00e7” noktalar\u0131d\u0131r.<\/p>\n
Herhangi bir d\u00fc\u011f\u00fcm ve o d\u00fc\u011f\u00fcm\u00fcn t\u00fcm \u00e7ocuklar\u0131, torunlar\u0131 vb. taraf\u0131ndan olu\u015fturulan bir alt a\u011fa\u00e7t\u0131r. Her d\u00fc\u011f\u00fcm, kendi alt a\u011fac\u0131n\u0131n k\u00f6k\u00fcd\u00fcr.<\/p>\n
\u0130ki d\u00fc\u011f\u00fcm aras\u0131ndaki ba\u011flant\u0131d\u0131r.<\/p>\n
Bir d\u00fc\u011f\u00fcmden di\u011ferine ula\u015fmak i\u00e7in izlenen ard\u0131\u015f\u0131k kenarlar\u0131n veya d\u00fc\u011f\u00fcmlerin dizisidir.<\/p>\n
Bir yoldaki kenar say\u0131s\u0131d\u0131r.<\/p>\n
K\u00f6k d\u00fc\u011f\u00fcmden belirli bir d\u00fc\u011f\u00fcme olan yolun uzunlu\u011fudur. K\u00f6k d\u00fc\u011f\u00fcm\u00fcn derinli\u011fi genellikle 0 olarak kabul edilir. \u00d6rne\u011fin, k\u00f6k\u00fcn \u00e7ocuklar\u0131n\u0131n derinli\u011fi 1, onlar\u0131n \u00e7ocuklar\u0131n\u0131n derinli\u011fi 2’dir ve bu b\u00f6yle devam eder.<\/p>\n
Derinlik ile e\u015f anlaml\u0131 olarak kullan\u0131labilir, ancak baz\u0131 ba\u011flamlarda k\u00f6k d\u00fc\u011f\u00fcm seviye 1 olarak da tan\u0131mlanabilir. Bu makalede, derinlik terimini kullan\u0131rken k\u00f6k d\u00fc\u011f\u00fcm\u00fcn derinli\u011fini 0 kabul edece\u011fiz, bu da seviye 0 anlam\u0131na gelir.<\/p>\n
A\u011fa\u00e7 y\u00fcksekli\u011fi, bir a\u011fac\u0131n “ne kadar derin” oldu\u011funu \u00f6l\u00e7en temel bir \u00f6zelliktir. Tan\u0131m\u0131, kullan\u0131lan literat\u00fcre ve ba\u011flama g\u00f6re biraz farkl\u0131l\u0131k g\u00f6sterebilse de, en yayg\u0131n iki tan\u0131m \u015funlard\u0131r:<\/p>\n
Bu tan\u0131ma g\u00f6re, bir a\u011fac\u0131n y\u00fcksekli\u011fi, k\u00f6k d\u00fc\u011f\u00fcmden herhangi bir yaprak d\u00fc\u011f\u00fcme giden en uzun yolun kenar say\u0131s\u0131d\u0131r.
\n* Bo\u015f bir a\u011fac\u0131n y\u00fcksekli\u011fi genellikle -1 olarak kabul edilir.
\n* Sadece k\u00f6k d\u00fc\u011f\u00fcmden olu\u015fan bir a\u011fac\u0131n (yani tek bir d\u00fc\u011f\u00fcm\u00fcn) y\u00fcksekli\u011fi 0’d\u0131r, \u00e7\u00fcnk\u00fc k\u00f6k ayn\u0131 zamanda yaprak d\u00fc\u011f\u00fcmd\u00fcr ve k\u00f6kten kendisine giden yolun uzunlu\u011fu 0’d\u0131r.
\n* K\u00f6k d\u00fc\u011f\u00fcmden yaprak d\u00fc\u011f\u00fcme giden en uzun yolun uzunlu\u011fu, o yaprak d\u00fc\u011f\u00fcm\u00fcn derinli\u011fine e\u015fittir. Dolay\u0131s\u0131yla, a\u011fa\u00e7 y\u00fcksekli\u011fi, t\u00fcm yaprak d\u00fc\u011f\u00fcmlerin derinliklerinin maksimumudur.<\/p>\n
Bu tan\u0131m, bilgisayar bilimlerinde ve algoritma analizinde en yayg\u0131n kullan\u0131lan\u0131d\u0131r ve bu makalede de bu tan\u0131m\u0131 esas alaca\u011f\u0131z.<\/p>\n
Baz\u0131 kaynaklar, a\u011fa\u00e7 y\u00fcksekli\u011fini k\u00f6k d\u00fc\u011f\u00fcmden herhangi bir yaprak d\u00fc\u011f\u00fcme giden en uzun yol \u00fczerindeki d\u00fc\u011f\u00fcm say\u0131s\u0131 olarak tan\u0131mlar.
\n* Bu tan\u0131ma g\u00f6re, bo\u015f bir a\u011fac\u0131n y\u00fcksekli\u011fi 0 olarak kabul edilebilir (veya tan\u0131ms\u0131z).
\n* Sadece k\u00f6k d\u00fc\u011f\u00fcmden olu\u015fan bir a\u011fac\u0131n y\u00fcksekli\u011fi 1’dir (tek d\u00fc\u011f\u00fcm).
\n* E\u011fer Tan\u0131m 1’i kullan\u0131rsak (kenar say\u0131s\u0131), bu tan\u0131m (d\u00fc\u011f\u00fcm say\u0131s\u0131) “y\u00fckseklik + 1” olur.<\/p>\n
Bu iki tan\u0131m aras\u0131ndaki fark\u0131 anlamak \u00f6nemlidir. \u00d6rne\u011fin, 3 d\u00fc\u011f\u00fcml\u00fc bir a\u011fa\u00e7ta (k\u00f6k -> \u00e7ocuk -> yaprak) Tan\u0131m 1’e g\u00f6re y\u00fckseklik 2 iken, Tan\u0131m 2’ye g\u00f6re y\u00fckseklik 3 olacakt\u0131r. Tutarl\u0131l\u0131k sa\u011flamak ad\u0131na, bu makalede “kenar say\u0131s\u0131” tan\u0131m\u0131n\u0131 (k\u00f6k derinli\u011fi 0 olan) benimseyece\u011fiz.<\/p>\n
Bir a\u011fac\u0131n y\u00fcksekli\u011fini hesaplamak i\u00e7in genellikle iki ana yakla\u015f\u0131m kullan\u0131l\u0131r: \u00f6zyinelemeli (recursive) ve yinelemeli (iterative). Her iki yakla\u015f\u0131m\u0131n da kendine \u00f6zg\u00fc avantajlar\u0131 ve dezavantajlar\u0131 vard\u0131r.<\/p>\n
\u00d6zyinelemeli yakla\u015f\u0131m, a\u011fa\u00e7 yap\u0131s\u0131n\u0131n do\u011fal \u00f6zyinelemeli tan\u0131m\u0131ndan yararlan\u0131r. Bir d\u00fc\u011f\u00fcm\u00fcn y\u00fcksekli\u011fi, o d\u00fc\u011f\u00fcm\u00fcn \u00e7ocuklar\u0131n\u0131n y\u00fcksekliklerinin maksimumunun bir fazlas\u0131d\u0131r.<\/p>\n
1. Temel Durum (Base Case):<\/strong> * Zaman Karma\u015f\u0131kl\u0131\u011f\u0131:<\/strong> Her d\u00fc\u011f\u00fcm tam olarak bir kez ziyaret edildi\u011fi i\u00e7in, \u00f6zyinelemeli yakla\u015f\u0131m\u0131n zaman karma\u015f\u0131kl\u0131\u011f\u0131 O(N)’dir, burada N a\u011fa\u00e7taki d\u00fc\u011f\u00fcm say\u0131s\u0131d\u0131r. Yinelemeli yakla\u015f\u0131m genellikle bir kuyruk (queue) kullanarak geni\u015flik \u00f6ncelikli arama (BFS) algoritmas\u0131 ile uygulan\u0131r. Bu y\u00f6ntem, a\u011fac\u0131 seviye seviye gezer ve her seviye ge\u00e7i\u015finde y\u00fcksekli\u011fi art\u0131r\u0131r.<\/p>\n 1. E\u011fer a\u011fa\u00e7 bo\u015fsa, y\u00fckseklik -1’dir. * Zaman Karma\u015f\u0131kl\u0131\u011f\u0131:<\/strong> Her d\u00fc\u011f\u00fcm yine tam olarak bir kez ziyaret edildi\u011fi i\u00e7in, zaman karma\u015f\u0131kl\u0131\u011f\u0131 O(N)’dir. A\u011fa\u00e7 y\u00fcksekli\u011fi, a\u011fac\u0131n yap\u0131s\u0131na ve t\u00fcr\u00fcne g\u00f6re \u00f6nemli \u00f6l\u00e7\u00fcde de\u011fi\u015fir. A\u011fa\u00e7 t\u00fcrleri, genellikle d\u00fc\u011f\u00fcmlerin nas\u0131l eklendi\u011fi, \u00e7\u0131kar\u0131ld\u0131\u011f\u0131 ve d\u00fczenlendi\u011fi konusunda belirli kurallara sahiptir. Bu kurallar, a\u011fac\u0131n y\u00fcksekli\u011fini ve dolay\u0131s\u0131yla performans\u0131n\u0131 do\u011frudan etkiler.<\/p>\n Her d\u00fc\u011f\u00fcm\u00fcn en fazla iki \u00e7ocu\u011fu olabilen a\u011fa\u00e7lard\u0131r (sol \u00e7ocuk ve sa\u011f \u00e7ocuk).<\/p>\n * Minimum Y\u00fckseklik:<\/strong> N d\u00fc\u011f\u00fcml\u00fc bir ikili a\u011fa\u00e7 i\u00e7in minimum y\u00fckseklik log\u2082(N+1) – 1’dir. Bu, m\u00fckemmel veya dolu ikili a\u011fa\u00e7lar i\u00e7in ge\u00e7erlidir. \u00d6rne\u011fin, 7 d\u00fc\u011f\u00fcml\u00fc m\u00fckemmel ikili a\u011fac\u0131n y\u00fcksekli\u011fi 2’dir (log\u2082(7+1) – 1 = 3 – 1 = 2). * Tam \u0130kili A\u011fa\u00e7 (Full Binary Tree):<\/strong> Her d\u00fc\u011f\u00fcm\u00fcn ya 0 ya da 2 \u00e7ocu\u011fu vard\u0131r. Y\u00fcksekli\u011fi Arama, ekleme ve silme i\u015flemlerinin O(log N) zaman karma\u015f\u0131kl\u0131\u011f\u0131na sahip olmas\u0131n\u0131 sa\u011flamak i\u00e7in tasarlanm\u0131\u015f a\u011fa\u00e7lard\u0131r. Bu a\u011fa\u00e7lar, d\u00fc\u011f\u00fcmlerin eklenmesi veya \u00e7\u0131kar\u0131lmas\u0131 s\u0131ras\u0131nda otomatik olarak kendi kendilerini dengelerler.<\/p>\n AVL A\u011fa\u00e7lar\u0131 (Adelson-Velsky and Landis Tree):<\/strong> Kendi kendini dengeleyen ilk ikili arama a\u011fac\u0131d\u0131r. Her d\u00fc\u011f\u00fcm i\u00e7in sol ve sa\u011f alt a\u011fa\u00e7lar\u0131n y\u00fckseklikleri aras\u0131ndaki fark en fazla 1 olabilir. Bu kat\u0131 dengeleme kural\u0131 sayesinde, N d\u00fc\u011f\u00fcml\u00fc bir AVL a\u011fac\u0131n\u0131n y\u00fcksekli\u011fi her zaman O(log N) olur. En k\u00f6t\u00fc durumda, y\u00fckseklik Her d\u00fc\u011f\u00fcm\u00fcn herhangi bir say\u0131da \u00e7ocu\u011fu olabilece\u011fi a\u011fa\u00e7lard\u0131r. Bu t\u00fcr a\u011fa\u00e7larda belirli bir y\u00fckseklik form\u00fcl\u00fc vermek zordur, \u00e7\u00fcnk\u00fc yap\u0131lar\u0131 \u00e7ok de\u011fi\u015fkendir. Y\u00fckseklik, a\u011fac\u0131n belirli anl\u0131k yap\u0131s\u0131na ba\u011fl\u0131d\u0131r ve yine O(N) ile O(log N) aras\u0131nda de\u011fi\u015febilir.<\/p>\n A\u011fa\u00e7 y\u00fcksekli\u011fi, bir a\u011fa\u00e7 veri yap\u0131s\u0131n\u0131n performans\u0131n\u0131 ve pratik kullan\u0131m\u0131n\u0131 do\u011frudan etkileyen kritik bir parametredir.<\/p>\n * Arama, Ekleme ve Silme \u0130\u015flemleri:<\/strong> A\u011fa\u00e7larda \u00e7o\u011fu temel i\u015flem (bir \u00f6\u011feyi arama, yeni bir \u00f6\u011fe ekleme, mevcut bir \u00f6\u011feyi silme) k\u00f6kten ba\u015flay\u0131p bir yaprak d\u00fc\u011f\u00fcme do\u011fru ilerlemeyi gerektirir. Bu i\u015flemlerin zaman karma\u015f\u0131kl\u0131\u011f\u0131, en k\u00f6t\u00fc durumda a\u011fac\u0131n y\u00fcksekli\u011fi ile orant\u0131l\u0131d\u0131r. * Veritaban\u0131 \u0130ndeksleme:<\/strong> B-A\u011fa\u00e7lar\u0131, veritaban\u0131 indekslerinde yayg\u0131n olarak kullan\u0131l\u0131r. Bir veritaban\u0131ndaki kay\u0131tlar\u0131 bulmak i\u00e7in diskten okunmas\u0131 gereken blok say\u0131s\u0131, B-a\u011fac\u0131n\u0131n y\u00fcksekli\u011fi ile orant\u0131l\u0131d\u0131r. Y\u00fcksekli\u011fin d\u00fc\u015f\u00fck tutulmas\u0131 (geni\u015f dallanma fakt\u00f6rleri ile), disk eri\u015fimini minimize ederek performans\u0131 art\u0131r\u0131r. * Dengeleme Algoritmalar\u0131:<\/strong> A\u011fa\u00e7 dengeleme algoritmalar\u0131 (rotasyonlar gibi), a\u011fac\u0131n y\u00fcksekli\u011fini belirli bir e\u015fik alt\u0131nda tutmay\u0131 hedefler. Bu algoritmalar, a\u011fac\u0131n genel performans\u0131n\u0131 optimize etmek i\u00e7in y\u00fckseklik bilgisini kullan\u0131r. A\u011fa\u00e7lar\u0131n g\u00f6rselle\u015ftirilmesinde, y\u00fckseklik bilgisi d\u00fczen ve \u00f6l\u00e7eklendirme i\u00e7in \u00f6nemlidir. Y\u00fckseklik, a\u011fac\u0131n ne kadar dikey alan kaplayaca\u011f\u0131n\u0131 belirler.<\/p>\n Y\u00fckseklik ve derinlik kavramlar\u0131 birbirleriyle yak\u0131ndan ili\u015fkilidir, ancak farkl\u0131 \u015feyleri \u00f6l\u00e7erler:<\/p>\n * D\u00fc\u011f\u00fcm Derinli\u011fi:<\/strong> K\u00f6k d\u00fc\u011f\u00fcmden belirli bir d\u00fc\u011f\u00fcme olan yolun uzunlu\u011fudur (kenar say\u0131s\u0131). K\u00f6k\u00fcn derinli\u011fi 0’d\u0131r. Ba\u015fka bir deyi\u015fle, a\u011fac\u0131n y\u00fcksekli\u011fi, a\u011fa\u00e7taki en derin d\u00fc\u011f\u00fcm\u00fcn derinli\u011fidir. Bu ili\u015fki, \u00f6zyinelemeli y\u00fckseklik hesaplama algoritmalar\u0131n\u0131n temelini olu\u015fturur; bir d\u00fc\u011f\u00fcm\u00fcn y\u00fcksekli\u011fi, o d\u00fc\u011f\u00fcm\u00fcn en derin alt a\u011fac\u0131n\u0131n y\u00fcksekli\u011fi art\u0131 kendisi i\u00e7in 1’dir.<\/p>\n A\u011fa\u00e7 veri yap\u0131lar\u0131nda y\u00fckseklik kavram\u0131, sadece teorik bir \u00f6l\u00e7\u00fc olman\u0131n \u00f6tesinde, bilgisayar bilimleri ve yaz\u0131l\u0131m m\u00fchendisli\u011finde pratik uygulamalar i\u00e7in hayati \u00f6neme sahiptir. A\u011fa\u00e7 y\u00fcksekli\u011fi, bir a\u011fac\u0131n ne kadar verimli \u00e7al\u0131\u015ft\u0131\u011f\u0131n\u0131, arama, ekleme ve silme gibi temel i\u015flemlerin ne kadar h\u0131zl\u0131 tamamlanaca\u011f\u0131n\u0131 do\u011frudan belirler. Dengeli a\u011fa\u00e7larda logaritmik y\u00fckseklik elde ederek performans\u0131 optimize etmek, b\u00fcy\u00fck veri k\u00fcmeleriyle \u00e7al\u0131\u015fan sistemler i\u00e7in kritik bir tasar\u0131m hedefidir.<\/p>\n Bo\u015f bir a\u011fa\u00e7 i\u00e7in -1, tek d\u00fc\u011f\u00fcml\u00fc bir a\u011fa\u00e7 i\u00e7in 0 olarak tan\u0131mlanan ve k\u00f6kten en uzak yapra\u011fa olan kenar say\u0131s\u0131 olarak \u00f6l\u00e7\u00fclen y\u00fckseklik, \u00f6zyinelemeli veya yinelemeli y\u00f6ntemlerle hesaplanabilir. Her iki y\u00f6ntemin de O(N) zaman karma\u015f\u0131kl\u0131\u011f\u0131 olsa da, alan karma\u015f\u0131kl\u0131\u011f\u0131 (s\u0131ras\u0131yla O(H) ve O(W)) a\u011fac\u0131n yap\u0131s\u0131na g\u00f6re farkl\u0131l\u0131k g\u00f6sterebilir.<\/p>\n AVL ve K\u0131rm\u0131z\u0131-Siyah a\u011fa\u00e7lar gibi kendi kendini dengeleyen a\u011fa\u00e7lar, y\u00fcksekliklerini logaritmik d\u00fczeyde tutarak garantili O(log N) performans sunarken, B-a\u011fa\u00e7lar\u0131 disk tabanl\u0131 sistemlerde I\/O i\u015flemlerini minimize etmek i\u00e7in d\u00fc\u015f\u00fck y\u00fcksekliklerini kullan\u0131r. A\u011fa\u00e7 y\u00fcksekli\u011finin bu derinlemesine anla\u015f\u0131lmas\u0131, daha verimli algoritmalar ve daha performansl\u0131 sistemler tasarlamak i\u00e7in temel bir ad\u0131md\u0131r. A\u011fa\u00e7lar\u0131n evrimi ve yeni veri yap\u0131lar\u0131n\u0131n geli\u015ftirilmesi devam ederken, y\u00fckseklik kavram\u0131 her zaman merkezi bir rol oynamaya devam edecektir.<\/body><\/p>\n","protected":false},"excerpt":{"rendered":"A\u011fa\u00e7 Veri Yap\u0131lar\u0131nda Y\u00fckseklik Kavram\u0131: Tan\u0131mlar, Hesaplama ve \u00d6nemi\nA\u011fa\u00e7lar, bilgisayar bilimlerinde ve yaz\u0131l\u0131m m\u00fchendisli\u011finde en te","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"csco_page_header_type":"","csco_page_load_nextpost":"","csco_page_subscribe_form":"","csco_page_contact_form":"","footnotes":""},"categories":[1],"tags":[],"class_list":{"0":"post-38930","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-genel","7":"cs-entry","8":"cs-video-wrap"},"yoast_head":"\n
\n * E\u011fer a\u011fa\u00e7 bo\u015fsa (yani k\u00f6k d\u00fc\u011f\u00fcm null<\/code> ise), y\u00fckseklik -1’dir.
\n * E\u011fer d\u00fc\u011f\u00fcm bir yaprak d\u00fc\u011f\u00fcmse (yani hi\u00e7 \u00e7ocu\u011fu yoksa), y\u00fcksekli\u011fi 0’d\u0131r.
\n2. \u00d6zyinelemeli Ad\u0131m:<\/strong>
\n * Bir d\u00fc\u011f\u00fcm\u00fcn y\u00fcksekli\u011fini bulmak i\u00e7in, \u00f6nce t\u00fcm \u00e7ocuk d\u00fc\u011f\u00fcmlerinin y\u00fcksekliklerini \u00f6zyinelemeli olarak hesaplay\u0131n.
\n * Bu \u00e7ocuk y\u00fcksekliklerinin en b\u00fcy\u00fc\u011f\u00fcn\u00fc bulun.
\n * D\u00fc\u011f\u00fcm\u00fcn y\u00fcksekli\u011fi, bulunan maksimum \u00e7ocuk y\u00fcksekli\u011finin bir fazlas\u0131d\u0131r.<\/p>\nPseudocode (\u0130kili A\u011fa\u00e7 i\u00e7in Basitle\u015ftirilmi\u015f):<\/h5>\n
function calculateHeight(node):\n if node is null:\n return -1 \/\/ Bo\u015f a\u011fa\u00e7 veya alt a\u011fa\u00e7\n \n \/\/ T\u00fcm \u00e7ocuk d\u00fc\u011f\u00fcmlerinin y\u00fcksekliklerini hesapla\n maxChildHeight = -1\n for each child in node.children:\n maxChildHeight = max(maxChildHeight, calculateHeight(child))\n \n return 1 + maxChildHeight \/\/ Kendi y\u00fcksekli\u011fi + en uzun alt a\u011fac\u0131n y\u00fcksekli\u011fi<\/code><\/pre>\nZaman ve Alan Karma\u015f\u0131kl\u0131\u011f\u0131:<\/h5>\n
\n* Alan Karma\u015f\u0131kl\u0131\u011f\u0131:<\/strong> \u00d6zyinelemeli \u00e7a\u011fr\u0131lar, \u00e7a\u011fr\u0131 y\u0131\u011f\u0131n\u0131nda (call stack) yer kaplar. En k\u00f6t\u00fc durumda (\u00e7arp\u0131k bir a\u011fa\u00e7ta), y\u0131\u011f\u0131n derinli\u011fi a\u011fac\u0131n y\u00fcksekli\u011fine (H) e\u015fit olabilir. Bu nedenle alan karma\u015f\u0131kl\u0131\u011f\u0131 O(H)’dir. Dengeli a\u011fa\u00e7larda H, log N’e yak\u0131nken, \u00e7arp\u0131k a\u011fa\u00e7larda N’e yak\u0131n olabilir.<\/p>\nYinelemeli Yakla\u015f\u0131m (BFS – Geni\u015flik \u00d6ncelikli Arama)<\/h4>\n
Algoritma Ad\u0131mlar\u0131:<\/h5>\n
\n2. K\u00f6k d\u00fc\u011f\u00fcm\u00fc bir kuyru\u011fa ekleyin.
\n3. Bir height<\/code> de\u011fi\u015fkenini -1 olarak ba\u015flat\u0131n.
\n4. Kuyruk bo\u015f olana kadar d\u00f6ng\u00fcy\u00fc s\u00fcrd\u00fcr\u00fcn:
\n * Mevcut seviyedeki d\u00fc\u011f\u00fcm say\u0131s\u0131n\u0131 al\u0131n (bu, kuyru\u011fun o anki boyutudur).
\n * height<\/code> de\u011ferini bir art\u0131r\u0131n.
\n * Mevcut seviyedeki t\u00fcm d\u00fc\u011f\u00fcmleri kuyruktan \u00e7\u0131kar\u0131n ve bu d\u00fc\u011f\u00fcmlerin \u00e7ocuklar\u0131n\u0131 kuyru\u011fa ekleyin.
\n * Bu i\u015flem, bir seviyedeki t\u00fcm d\u00fc\u011f\u00fcmlerin i\u015flendi\u011fini ve bir sonraki seviyenin d\u00fc\u011f\u00fcmlerinin kuyru\u011fa eklendi\u011fini garanti eder.
\n5. D\u00f6ng\u00fc bitti\u011finde, height<\/code> de\u011fi\u015fkeni a\u011fac\u0131n y\u00fcksekli\u011fini i\u00e7erecektir.<\/p>\nPseudocode (\u0130kili A\u011fa\u00e7 i\u00e7in Basitle\u015ftirilmi\u015f):<\/h5>\n
function calculateHeightIterative(root):\n if root is null:\n return -1\n \n queue = new Queue()\n queue.enqueue(root)\n height = -1 \/\/ K\u00f6k derinli\u011fi 0 oldu\u011fundan, ilk seviye i\u00e7in 0'a ula\u015fmak i\u00e7in -1'den ba\u015flar\u0131z\n \n while not queue.isEmpty():\n levelSize = queue.size()\n height = height + 1\n \n for i from 0 to levelSize - 1:\n currentNode = queue.dequeue()\n \n if currentNode.left is not null:\n queue.enqueue(currentNode.left)\n if currentNode.right is not null:\n queue.enqueue(currentNode.right)\n \n return height<\/code><\/pre>\nZaman ve Alan Karma\u015f\u0131kl\u0131\u011f\u0131:<\/h5>\n
\n* Alan Karma\u015f\u0131kl\u0131\u011f\u0131:<\/strong> Kuyruk, bir seviyedeki en fazla d\u00fc\u011f\u00fcm\u00fc saklamak zorunda kalabilir. En k\u00f6t\u00fc durumda (dengeli bir a\u011fa\u00e7ta son seviye), bu O(W) olabilir, burada W a\u011fac\u0131n en geni\u015f seviyesindeki d\u00fc\u011f\u00fcm say\u0131s\u0131d\u0131r. Tam bir ikili a\u011fa\u00e7ta W, N\/2’ye yak\u0131n olabilir, dolay\u0131s\u0131yla O(N) olarak da ifade edilebilir. Ancak, \u00f6zyinelemeli yakla\u015f\u0131m\u0131n y\u0131\u011f\u0131n derinli\u011fi O(H) iken, yinelemeli yakla\u015f\u0131m O(W) alan kullan\u0131r ki bu \u00f6zellikle \u00e7arp\u0131k a\u011fa\u00e7lar i\u00e7in daha iyi olabilir (\u00e7arp\u0131k a\u011fa\u00e7ta W=1).<\/p>\nFarkl\u0131 A\u011fa\u00e7 T\u00fcrlerinde Y\u00fckseklik<\/h3>\n
\u0130kili A\u011fa\u00e7lar (Binary Trees)<\/h4>\n
\n* Maksimum Y\u00fckseklik:<\/strong> N d\u00fc\u011f\u00fcml\u00fc bir ikili a\u011fa\u00e7 i\u00e7in maksimum y\u00fckseklik N-1’dir. Bu durum, “\u00e7arp\u0131k a\u011fa\u00e7” (skewed tree) olarak bilinen, her d\u00fc\u011f\u00fcm\u00fcn sadece bir \u00e7ocu\u011fu oldu\u011fu (sol veya sa\u011f) bir a\u011fa\u00e7ta meydana gelir. Bu durumda a\u011fa\u00e7 bir ba\u011flant\u0131l\u0131 listeye benzer.<\/p>\n\u00d6zel \u0130kili A\u011fa\u00e7 T\u00fcrleri:<\/h5>\n
N-1<\/code> (\u00e7arp\u0131k) ile log\u2082(N+1)-1<\/code> (dengeli) aras\u0131nda de\u011fi\u015fir.
\n* Dolu \u0130kili A\u011fa\u00e7 (Complete Binary Tree):<\/strong> Son seviye hari\u00e7 t\u00fcm seviyeler tamamen doludur ve son seviyedeki d\u00fc\u011f\u00fcmler m\u00fcmk\u00fcn oldu\u011funca sola yaslanm\u0131\u015ft\u0131r. Bu a\u011fa\u00e7lar her zaman dengelidir ve y\u00fckseklikleri her zaman O(log N) civar\u0131ndad\u0131r.
\n* M\u00fckemmel \u0130kili A\u011fa\u00e7 (Perfect Binary Tree):<\/strong> T\u00fcm i\u00e7 d\u00fc\u011f\u00fcmlerin iki \u00e7ocu\u011fu vard\u0131r ve t\u00fcm yaprak d\u00fc\u011f\u00fcmler ayn\u0131 derinliktedir. Bu a\u011fa\u00e7lar da her zaman dengelidir ve y\u00fckseklikleri O(log N)’dir. N d\u00fc\u011f\u00fcml\u00fc m\u00fckemmel ikili a\u011fac\u0131n y\u00fcksekli\u011fi log\u2082(N+1) - 1<\/code>‘dir.<\/p>\nDengeli A\u011fa\u00e7lar (Balanced Trees)<\/h4>\n
1.44 <\/em> log\u2082(N+2)<\/code> civar\u0131ndad\u0131r.
\n K\u0131rm\u0131z\u0131-Siyah A\u011fa\u00e7lar (Red-Black Trees):<\/strong> AVL a\u011fa\u00e7lar\u0131na g\u00f6re biraz daha az kat\u0131 dengeleme kurallar\u0131na sahip kendi kendini dengeleyen ikili arama a\u011fa\u00e7lar\u0131d\u0131r. Her d\u00fc\u011f\u00fcm k\u0131rm\u0131z\u0131 veya siyaht\u0131r ve belirli kurallara uymak zorundad\u0131rlar (\u00f6rne\u011fin, k\u0131rm\u0131z\u0131 d\u00fc\u011f\u00fcm\u00fcn \u00e7ocuklar\u0131 siyah olmal\u0131d\u0131r). N d\u00fc\u011f\u00fcml\u00fc bir K\u0131rm\u0131z\u0131-Siyah a\u011fac\u0131n\u0131n y\u00fcksekli\u011fi de O(log N)’dir. En k\u00f6t\u00fc durumda, y\u00fckseklik 2 <\/em> log\u2082(N+1)<\/code>‘den fazla de\u011fildir.
\n* B-A\u011fa\u00e7lar\u0131 (B-Trees):<\/strong> Genellikle veritabanlar\u0131 ve dosya sistemleri gibi disk tabanl\u0131 veri yap\u0131lar\u0131 i\u00e7in kullan\u0131lan \u00e7ok yollu arama a\u011fa\u00e7lar\u0131d\u0131r. Her d\u00fc\u011f\u00fcm\u00fcn birden fazla \u00e7ocu\u011fu olabilir (genellikle bir ‘m’ de\u011feri ile tan\u0131mlan\u0131r). B-a\u011fa\u00e7lar\u0131n\u0131n y\u00fcksekli\u011fi, dallanma fakt\u00f6r\u00fc ‘m’ ile ili\u015fkili olarak log_m N<\/code>‘dir. Disk eri\u015fim maliyetini minimize etmek i\u00e7in y\u00fckseklik m\u00fcmk\u00fcn oldu\u011funca d\u00fc\u015f\u00fck tutulur.<\/p>\nGenel A\u011fa\u00e7lar (General Trees)<\/h4>\n
A\u011fa\u00e7 Y\u00fcksekli\u011finin \u00d6nemi ve Uygulamalar\u0131<\/h3>\n
Performans Analizi<\/h4>\n
\n * Dengeli A\u011fa\u00e7lar:<\/strong> AVL veya K\u0131rm\u0131z\u0131-Siyah a\u011fa\u00e7lar gibi dengeli a\u011fa\u00e7larda y\u00fckseklik O(log N) oldu\u011fu i\u00e7in, bu i\u015flemlerin zaman karma\u015f\u0131kl\u0131\u011f\u0131 da O(log N)’dir. Bu, \u00e7ok b\u00fcy\u00fck veri k\u00fcmeleri i\u00e7in bile h\u0131zl\u0131 eri\u015fim ve de\u011fi\u015fiklik anlam\u0131na gelir.
\n * Dengesiz A\u011fa\u00e7lar (\u00c7arp\u0131k A\u011fa\u00e7lar):<\/strong> En k\u00f6t\u00fc durumda, a\u011fa\u00e7 \u00e7arp\u0131k hale geldi\u011finde (y\u00fckseklik O(N)), arama, ekleme ve silme i\u015flemleri O(N) zaman alabilir. Bu, b\u00fcy\u00fck veri k\u00fcmeleri i\u00e7in kabul edilemez derecede yava\u015f olabilir ve a\u011fac\u0131n bir ba\u011flant\u0131l\u0131 liste gibi davranmas\u0131na neden olur.
\n* Bellek Kullan\u0131m\u0131:<\/strong> \u00d6zyinelemeli algoritmalar kullan\u0131ld\u0131\u011f\u0131nda, \u00e7a\u011fr\u0131 y\u0131\u011f\u0131n\u0131n\u0131n derinli\u011fi a\u011fac\u0131n y\u00fcksekli\u011fine ba\u011fl\u0131d\u0131r. \u00c7ok y\u00fcksek a\u011fa\u00e7lar, y\u0131\u011f\u0131n ta\u015fmas\u0131 (stack overflow) riskini art\u0131rabilir.<\/p>\nVeri Eri\u015fimi ve Depolama<\/h4>\n
\n* Dosya Sistemleri:<\/strong> Modern dosya sistemleri (\u00f6rne\u011fin NTFS, HFS+) genellikle B-a\u011fa\u00e7lar\u0131n\u0131n veya t\u00fcrevlerinin (B+ a\u011fa\u00e7lar\u0131) varyantlar\u0131n\u0131 kullanarak dosya ve dizin yap\u0131lar\u0131n\u0131 y\u00f6netir. Burada da y\u00fckseklik, dosyalara eri\u015fim s\u00fcresini belirleyen ana fakt\u00f6rlerden biridir.<\/p>\nAlgoritma Tasar\u0131m\u0131<\/h4>\n
\n* Minimum Spanning Tree (MST) Algoritmalar\u0131:<\/strong> Baz\u0131 grafik algoritmalar\u0131nda (\u00f6rne\u011fin Kruskal’\u0131n algoritmas\u0131), birle\u015fim-bulma (union-find) veri yap\u0131s\u0131 kullan\u0131l\u0131r ve bu veri yap\u0131s\u0131 a\u011fa\u00e7 tabanl\u0131d\u0131r. Bu a\u011fa\u00e7lar\u0131n y\u00fcksekli\u011fi, birle\u015fim-bulma i\u015flemlerinin amortize edilmi\u015f zaman karma\u015f\u0131kl\u0131\u011f\u0131n\u0131 etkiler.<\/p>\nGrafiksel Temsil ve G\u00f6rselle\u015ftirme<\/h4>\n
Y\u00fckseklik ve Derinlik Aras\u0131ndaki \u0130li\u015fki<\/h3>\n
\n* A\u011fa\u00e7 Y\u00fcksekli\u011fi:<\/strong> K\u00f6kten bir yaprak d\u00fc\u011f\u00fcme olan en uzun yolun uzunlu\u011fudur. Bu, a\u011fa\u00e7taki t\u00fcm yaprak d\u00fc\u011f\u00fcmlerin derinliklerinin maksimum de\u011feridir.<\/p>\nSonu\u00e7<\/h3>\n