# AVL TREE ROTATIONS PDF

AVL tree is a self-balancing Binary Search Tree BST where the difference between heights of left and right subtrees cannot be more than one for all nodes. Why AVL Trees? Most of the BST operations e. The cost of these operations may become O n for a skewed Binary tree. If we make sure that height of the tree remains O Logn after every insertion and deletion, then we can guarantee an upper bound of O Logn for all these operations. The height of an AVL tree is always O Logn where n is the number of nodes in the tree See this video lecture for proof.

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AVL tree is a self-balancing Binary Search Tree BST where the difference between heights of left and right subtrees cannot be more than one for all nodes. Why AVL Trees? Most of the BST operations e.

The cost of these operations may become O n for a skewed Binary tree. If we make sure that height of the tree remains O Logn after every insertion and deletion, then we can guarantee an upper bound of O Logn for all these operations.

The height of an AVL tree is always O Logn where n is the number of nodes in the tree See this video lecture for proof. Insertion To make sure that the given tree remains AVL after every insertion, we must augment the standard BST insert operation to perform some re-balancing.

Let z be the first unbalanced node, y be the child of z that comes on the path from w to z and x be the grandchild of z that comes on the path from w to z. There can be 4 possible cases that needs to be handled as x, y and z can be arranged in 4 ways.

Following are the possible 4 arrangements: a y is left child of z and x is left child of y Left Left Case b y is left child of z and x is right child of y Left Right Case c y is right child of z and x is right child of y Right Right Case d y is right child of z and x is left child of y Right Left Case. Following are the operations to be performed in above mentioned 4 cases. In all of the cases, we only need to re-balance the subtree rooted with z and the complete tree becomes balanced as the height of subtree After appropriate rotations rooted with z becomes same as it was before insertion.

See this video lecture for proof. Insertion Examples:. The following implementation uses the recursive BST insert to insert a new node. In the recursive BST insert, after insertion, we get pointers to all ancestors one by one in a bottom-up manner.

The recursive code itself travels up and visits all the ancestors of the newly inserted node. Update the height of the current node. To check whether it is left left case or not, compare the newly inserted key with the key in left subtree root. To check whether it is Right Right case or not, compare the newly inserted key with the key in right subtree root.

Time Complexity: The rotation operations left and right rotate take constant time as only a few pointers are being changed there. Updating the height and getting the balance factor also takes constant time. The AVL trees are more balanced compared to Red-Black Trees, but they may cause more rotations during insertion and deletion.

So if your application involves many frequent insertions and deletions, then Red Black trees should be preferred.

And if the insertions and deletions are less frequent and search is the more frequent operation, then AVL tree should be preferred over Red Black Tree. Following is the post for delete. Median in a stream of integers running integers Maximum of all subarrays of size k Count smaller elements on right side. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. GeeksforGeeks has prepared a complete interview preparation course with premium videos, theory, practice problems, TA support and many more features.

Please refer Placement for details. Writing code in comment? Please use ide. NULL left and right pointers. Get the balance factor of this ancestor. Node left, right;. Node root;. Python code to insert a node in AVL tree. AVL tree class which supports the. Recursive function to insert key in.

Step 1 - Perform normal BST. Step 2 - Update the height of the. Step 3 - Get the balance factor. Step 4 - If the node is unbalanced,. Case 1 - Left Left. Case 2 - Right Right. Case 3 - Left Right. Case 4 - Right Left. Perform rotation. Update heights. Return the new root. Driver program to test above function. This code is contributed by Ajitesh Pathak. Node rightRotate Node y. Node leftRotate Node x. Node insert Node node, int key.

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## Data Structure and Algorithms - AVL Trees

An AVL tree is a subtype of binary search tree. Named after it's inventors Adelson, Velskii and Landis, AVL trees have the property of dynamic self-balancing in addition to all the properties exhibited by binary search trees. You will do an insertion similar to a normal Binary Search Tree insertion. After inserting, you fix the AVL property using left or right rotations. An AVL tree is a self-balancing binary search tree. AVL tree checks the height of the left and the right sub-trees and assures that the difference is not more than 1. This difference is called the Balance Factor.

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## AVL Tree | Set 1 (Insertion)

What if the input to binary search tree comes in a sorted ascending or descending manner? In real-time data, we cannot predict data pattern and their frequencies. So, a need arises to balance out the existing BST. AVL tree checks the height of the left and the right sub-trees and assures that the difference is not more than 1. This difference is called the Balance Factor. In the second tree, the left subtree of C has height 2 and the right subtree has height 0, so the difference is 2.

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## AVL Tree Insertion, Rotation, and Balance Factor Explained

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