Binary Search Trees Lecture 6 Asst. Prof. Dr. İlker Kocabaş 1

Binary search tree sort 2

Binary-search-tree property 3

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. 4

Binary-search-tree property 5

Inorder-tree walk During this type of walk, we visit the root of a subtree between the left subtree visit and right subtree visit. 6

Preorder-tree walk In this case we visit the root node before the nodes in either subtree. 7

Postorder-tree walk We visit the root node after the nodes in its subtrees. 8

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 10

Sorting by binary-search-tree (contnd.) 11

Searching for a key Search for 13 12

Searching for a key 13

Searching for minimum Minimum 14

Searching for minimum 15

Searching for maximum Maximum 16

Searching for maximum 17

Searching for successor The successor of a node x is the node with the smallest key greather than key [ x ] 18

Searching for successor Successor of 15 Case 1: The right subtree of node x is nonempty 19

Searching for successor Successor of 13 Case 2: The right subtree of node x is empty 20

Searching for successor Successor of 4 search until y.left = x Case 2: The right subtree of node x is empty 21

Searching for successor 22  Go upper until y.left = x

Insertion Inserting an item with key 13 23

Insertion 24

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 (6 successor of 5) (z has two children) 10 z y

Deletion Deleting a node with key 5 (z has two children) 10 z y 7 30

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

Deletion 32

Analysis of BST sort The expected time to built the tree is asymptotically the same as the running time of quicksort 33

Red Black Trees 34

Balanced search trees 35

Red-black trees 36

Example of a red-black tree 37

Example of a red-black tree 38

Example of a red-black tree 39

Example of a red-black tree 40

Example of a red-black tree 41

Height of a red-black tree 42

Height of a red-black tree 43

Height of a red-black tree 44

Height of a red-black tree 45

Height of a red-black tree 46

Height of a red-black tree 47

Proof (continued) 48

Query operations 49

Modifying operations 50

Rotations 51

Insertion into a red-black tree 52

Insertion into a red-black tree 53

Insertion into a red-black tree 54

Insertion into a red-black tree 55

Insertion into a red-black tree 56

Pseudocode 57

Graphical notation 58

Case 1 59

Case 2 60

Case 3 61

Analysis 62