Binary Search Trees Lecture 6 Prof. Dr. Aydın Öztürk

Binary search tree sort

Binary-search-tree property

Binary Search Tree can be implemented as a linked data structure in which each node is an object with three pointer fields. The three pointer fields left, right and p point to the nodes corresponding to the left child, right child and the parent respectively. NIL in any pointer field signifies that there exists no corresponding child or parent. The root node is the only node in the BTS structure with NIL in its p field.

Binary-search-tree property

Inorder-tree walk During this type of walk, we visit the root of a subtree between the left subtree visit and right subtree visit.

Preorder-tree walk In this case we visit the root node before the nodes in either subtree.

Preorder-tree walk We visit the root node after the nodes in its subtrees.

Sorting by binary-search-tree NIL 3PRINT 2 4 NIL 3 PRINT NIL 3PRINT 5 4NIL 2 1 2NIL 3PRINT 8 4NIL 5 3 PRINT NIL 3PRINT

Sorting by binary-search-tree

Sorting by binary-search- tree(contnd.)

Searching for a key Search for 13

Searching for a key

Searching for minimum Minimum

Searching for minimum

Searching for maximum Maximum

Searching for maximum

Searching for successor The successor of a node x is the node with the smallest key greather than key [ x ]

Searching for successor Successor of 15 Case 1: The right subtree of node x is nonempty

Searching for successor Successor of 13 Case 2: The right subtree of node x is empty

Searching for successor

Insertion Insering an item with key 13

Insertion

Deletion Deleting an item with key 13 (z has no children) z

Deletion Deleting an item with key 13 (z has no children)

Deletion z Deleting a node with key 16 (z has only one child )

Deletion Deleting a node with key 16 (z has only one child )

Deletion Deleting a node with key 5 (z has two children) 10 z y 7 13

Deletion Deleting a node with key 5 (z has two children) 10 z y 7

Deletion Deleting a node with key 5 (z has two children)

Deletion

Analysis of BST sort The expected time to built the tree is asymptotically the same as the running time of quicksort

Hight of a randomly built binary search tree

Convex functions

Jensens’s inequality Proof: We assume that f has a Taylor expansion with µ=E(X).

Analysis of BST height

Analysis(contd.)

Exponential height recurrence

Solving the recurrence

Solving the recurrence (contnd.)