Editing Binary search tree (section)

===Searching===
Searching in a binary search tree for a specific key can be programmed [[recursion (computer science)|recursively]] or [[iteration#Computing|iteratively]].

Searching begins by examining the [[root node]]. If the tree is [[Null pointer|{{math|{{text|nil}}}}]], the key being searched for does not exist in the tree. Otherwise, if the key equals that of the root, the search is successful and the node is returned. If the key is less than that of the root, the search proceeds by examining the left subtree. Similarly, if the key is greater than that of the root, the search proceeds by examining the right subtree. This process is repeated until the key is found or the remaining subtree is <math>\text{nil}</math>. If the searched key is not found after a <math>\text{nil}</math> subtree is reached, then the key is not present in the tree.{{r|algo_cormen|pp=290-291}}

====Recursive search====
The following [[pseudocode]] implements the BST search procedure through [[recursion (computer science)|recursion]].<ref name="algo_cormen" />{{rp|290}}
{|
|- style="vertical-align:top"
|
 Recursive-Tree-Search(x, key)
     '''if''' x = NIL '''or''' key = x.key '''then'''
         '''return''' x
     '''if''' key < x.key '''then'''
         '''return''' Recursive-Tree-Search(x.left, key)
     '''else'''
         '''return''' Recursive-Tree-Search(x.right, key)
     '''end if'''
|}
The recursive procedure continues until a <math>\text{nil}</math> or the <math>\text{key}</math> being searched for are encountered.

====Iterative search====
The recursive version of the search can be "unrolled" into a [[while loop]]. On most machines, the iterative version is found to be more [[Computer performance|efficient]].<ref name="algo_cormen" />{{rp|291}}
{|
|- style="vertical-align:top"
|
 Iterative-Tree-Search(x, key)
     '''while''' x &ne; NIL '''and''' key &ne; x.key '''do'''
         '''if''' key < x.key '''then'''
             x := x.left
         '''else'''
             x := x.right
         '''end if'''
     '''repeat'''
     '''return''' x
|}
Since the search may proceed till some [[leaf node]], the running time complexity of BST search is <math>O(h)</math> where <math>h</math> is the [[Tree (data structure)#Terminology|height of the tree]]. However, the worst case for BST search is <math>O(n)</math> where <math>n</math> is the total number of nodes in the BST, because an unbalanced BST may degenerate to a linked list. However, if the BST is [[height-balanced tree|height-balanced]] the height is <math>O(\log n)</math>.<ref name="algo_cormen" />{{rp|290}}

====Successor and predecessor====
For certain operations, given a node <math>\text{x}</math>, finding the successor or predecessor of <math>\text{x}</math> is crucial. Assuming all the keys of a BST are distinct, the successor of a node <math>\text{x}</math> in a BST is the node with the smallest key greater than <math>\text{x}</math>'s key. On the other hand, the predecessor of a node <math>\text{x}</math> in a BST is the node with the largest key smaller than <math>\text{x}</math>'s key. The following pseudocode finds the successor and predecessor of a node <math>\text{x}</math> in a BST.<ref>{{cite web|url=https://ranger.uta.edu/~huang/teaching/CSE5311/CSE5311_Lecture10.pdf|archive-url=https://web.archive.org/web/20210413045057/http://ranger.uta.edu/~huang/teaching/CSE5311/CSE5311_Lecture10.pdf|archive-date=13 April 2021|page=12|publisher=[[University of Texas at Arlington]]|access-date=17 May 2021|url-status=live|title=Design and Analysis of Algorithms|author=Junzhou Huang}}</ref><ref>{{cite web |author=Ray |first=Ray |title=Binary Search Tree |url=https://cs.lmu.edu/~ray/notes/binarysearchtrees/ |access-date=17 May 2022 |publisher=[[Loyola Marymount University]], Department of Computer Science}}</ref><ref name="algo_cormen" />{{rp|292–293}}
{|
|- style="vertical-align:top"
|
  BST-Successor(x)
      '''if''' x.right &ne; NIL '''then'''
          '''return''' BST-Minimum(x.right)
      '''end if'''
      y := x.parent
      '''while''' y &ne; NIL '''and''' x = y.right '''do'''
          x := y
          y := y.parent
      '''repeat'''
      '''return''' y
|
  BST-Predecessor(x)
      '''if''' x.left &ne; NIL '''then'''
          '''return''' BST-Maximum(x.left)
      '''end if'''
      y := x.parent
      '''while''' y &ne; NIL '''and''' x = y.left '''do'''
          x := y
          y := y.parent
      '''repeat'''
      '''return''' y
|}
Operations such as finding a node in a BST whose key is the maximum or minimum are critical in certain operations, such as determining the successor and predecessor of nodes. Following is the pseudocode for the operations.<ref name="algo_cormen" />{{rp|291–292}}
{|
|- style="vertical-align:top"
|
  BST-Maximum(x)
      '''while''' x.right &ne; NIL '''do'''
          x := x.right
      '''repeat'''
      '''return''' x
|
  BST-Minimum(x)
      '''while''' x.left &ne; NIL '''do'''
          x := x.left
      '''repeat'''
      '''return''' x
|}