AVL Tree Chapter 6 (cont’)

Search Trees Two standard search trees: Binary Search Trees (non-balanced) All items in left sub-tree are less than root All items in right sub-tree are greater than or equal to the root Each sub-tree is a binary search tree AVL trees (balanced)

1- Binary Search Trees(review)

BST Traversals A preorder traversal of the BST produces: 23 18 12 20 44 35 52 A postorder traversal of the BST produces: 12 20 18 35 52 44 23 An inorder traversal of the BST produces: 12 18 20 23 35 44 52 (a sorted list!)

BST Search Algorithms Find Smallest Node: recursively follow the left branch until reaching a leaf Find Largest Node: recursively follow the right branch until reaching a leaf BST Search: recursively search for key node in a sub-tree (similar to the binary search)

Algorithm : Find Smallest algorithm findSmallestBST (val root <pointer>) if (root->left null) return (root) end if return findSmallestBST (root->left) end findSmallestBST

Algorithm : Find Largest algorithm findLargestBST(val root <pointer>) if (root->right null) return (root) end if return findLargestBST(root->right) end findLargestBST

Algorithm : BST Search algorithm searchBST (ref root <pointer>, val argument <key>) if (root is null) return null end if if (argument < root->key) return searchBST (root->left, argument) elseif (argument > root->key) return searchBST (root->right, argument) else return root end searchBST

Algorithm : Recursively add node to BST algorithm addBST (ref root <pointer>, val new <pointer>) if (root is null) root = new else Locate null sub-tree for insertion if (new->key < root->key) addBst (root->left, new) addBst (root->right, new) end if return end addBST

Insert

Insert

Delete

Delete

2- AVL Trees Invented by Adelson-Velskii and Landis Height-balanced binary search tree where the heights of sub-trees differ by no more than 1: | HL – HR | <= 1 Search effort for an AVL tree is O(log2n) Each node has a balance factor Balance factors may be: Left High (LH) = +1 (left sub-tree higher than right sub-tree) Even High (EH) = 0 (left and right sub-trees same height) Right High (RH) = -1 (right sub-tree higher than left sub-tree)

Figure 8-12

Figure 8-13 An AVL tree

Balancing Trees Insertions and deletions potentially cause a tree to be imbalanced When a tree is detected as unbalanced, nodes are balanced by rotating nodes to the left or right Four imbalance cases: Left of left: the sub-tree of a left high tree has become left high Right of right: the sub-tree of a right high tree has become right high Right of left: the sub-tree of a left high tree has become right high Left of right: the sub-tree of a right high tree has become left high Each imbalance case has simple and complex rotation cases

Case 1: Left of Left The sub-tree of a left high tree has become left high Simple right rotation: Rotate the out of balance node (the root) to the right Complex right rotation: Rotate root to the right, so the old root is the right sub-tree of the new root; the new root's right sub-tree is connected to the old root's left sub-tree

Case 2: Right of Right Mirror of Case 1: The sub-tree of a right high tree has become right high Simple left rotation: Rotate the out of balance node (the root) to the left Complex left rotation: Rotate root to the left, so the old root is the left sub-tree of the new root; the new root's left sub-tree is connected to the old root's right sub-tree

Case 3: Right of Left The sub-tree of a left high tree has become right high Simple double rotation right: Rotate left sub-tree to the left; rotate root to the right, so the old root's left node is the new root Complex double rotation right: Rotate the right-high left sub-tree to the left; rotate the left-high left sub-tree to the right

Case 4: Left of Right Mirror of Case 3: The sub-tree of a right high tree has become left high Simple double rotation right: Rotate right sub-tree to the right; rotate root to the left, so the old root's right node is the new root Complex double rotation right: Rotate the left-high right sub-tree to the right; rotate the right-high right sub-tree to the left

AVL Node Structure Node key <keyType> data <dataType> Left <pointer to Node> right <pointer to Node> bal <LH, EH, RH> // Balance factor End Node