Deletion may disturb the balance factor of an AVL tree and therefore the tree needs to be rebalanced in order to maintain the AVLness. Deletion Operation in AVL Tree. Learn about AVL trees and its insertion and deletion. Proof of its height and code in C, Java and Python. • insert, delete can be implemented in O(log n) time. Red-black properties: 1. But after every deletion operation, we need to check with the Balance Factor condition. The deletion operation in AVL Tree is similar to deletion operation in BST. An AVL tree is 1. height-balanced 2. a Binary search tree We saw that the maximum height of an AVL tree with N nodes is O(log n). Deletion in AVL Tree. Properties & Notes • All leaves have balance = 0 • AVL tree with n nodes has height O(log n). Output: Preorder traversal of the constructed AVL tree is 9 1 0 -1 5 2 6 10 11 Preorder traversal after deletion of 10 1 0 -1 9 5 2 6 11 Time Complexity: The rotation operations (left and right rotate) take constant time as only few pointers are being changed there. insert, search, delete, maximum, minimum, predecessor and successor. In this example, consider the value 12 getting deleted: 10 / \ 5 12 (delete this node) \ 8 In this instance, after the nod storing 12 is deleted, we move up to the parent, 10. ⇒ ﬁnd will run in O(log n) time if AVL has binary search tree property. We will try to understand this algorithm using an example but before that let's go over the major steps of this algorithm. Learn about AVL trees and its insertion and deletion. This algorithm is similar to AVL insertion algorithm when it comes to height balancing. Updating the height and getting the balance factor also take constant time. 2. Identifying which node to delete can be tricky if the key is at an internal node. The root and leaves (NIL ’s) are black. For this purpose, we need to perform rotations. AVL Tree Delete Examples 1) The most simple example is formed when a node from a tree with four nodes gets deleted. Deleting a node from an AVL tree is similar to that in a binary search tree. AVL Trees 12 AVL Tree • An AVL Tree is a binary search tree such that for every internal node v of T, the heights of the children of v can differ by at most 1. If that is true, then find, insert, and remove, will all be O(Log N). We want to show that after an insertion or deletion (also O(log n) since the height is O(log n)), we can rebalance the tree in O(log n) time. Node Deletion Deletion of a node from an AVL Tree proceeds in exactly the same manner as in an arbitrary binary search tree. Recall that the idea is based on identifying thesuccessor node. As with insertion, additional steps must be taken to maintain balance factors and tree admissibility. AVL tree deletion algorithm is basically a modification of BST deletion algorithm. Proof of its height and code in C, Java and Python ... Let's look at some examples of deletion and then the reason for the same will be clear. L10.3 . Red-black trees . Examples: •AVL trees •2-3 trees •2-3-4 trees •B-trees •Red-black trees. • An example of an AVL tree where the heights are shown next to the nodes: 88 44 17 78 32 50 48 62 2 4 1 1 2 3 1 1 This data structure requires an extra one-bit color field in each node. Suppose that the node holding the key we want to remove is an internal node named u. The two types of rotations are L rotation and R rotation. Every node is either red or black. Example: Construct an AVL Tree by inserting numbers from 1 to 8. Deletion This example of deletion happened to be easy because the node holding the key to be removed was aleafnode. If a node is red, then its parent is black. AV L Height Lemma: The height of an AVL tree storing n keys is O(logn) Example of AVL: Question 1 A node in a binary tree is an only-child if it has a parent node but no sibling node (Note: The root does not qualify as an only child). 3. ⇒ Good structure to implement dictionary or sorted set ADTs. (The task of node deletion can always be reduced to that of deleting a node that has at most one child.)

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