An AVL tree is a self-balancing binary search tree. It was named after its inventors, Adelson-Velskii and Landis.
The AVL tree maintains a balance factor for each node, which is the difference between the heights of its left and right subtrees. This balance factor ensures that the tree remains balanced, i.e., the height difference between the left and right subtrees of any node is at most one.
Why Use an AVL Tree?
AVL trees are particularly useful when you need efficient operations for searching, insertion, and deletion in a sorted data structure. Unlike regular binary search trees, which can become highly imbalanced in certain scenarios (leading to poor performance), AVL trees ensure that the height of the tree remains logarithmic.
By maintaining balance, AVL trees guarantee an average time complexity of O(log n) for basic operations like searching, insertion, and deletion. This makes them ideal for applications where fast access to sorted data is essential.
How Does an AVL Tree Work?
An AVL tree follows the same principles as a binary search tree with some additional rules to maintain balance:
- Left Rotation: When the right subtree is larger than the left subtree by more than one level, a left rotation is performed to restore balance.
- Right Rotation: When the left subtree is larger than the right subtree by more than one level, a right rotation is performed to restore balance.
The rotations are performed recursively up through the tree until balance is restored.
To insert a new node into an AVL tree:
- Add as you would in any binary search tree based on comparison with existing nodes.
- Update the balance factor of each node along the path from the newly inserted node to the root.
- If the balance factor of any node becomes greater than one or less than negative one, perform appropriate rotations to restore balance.
To delete a node from an AVL tree:
- Remove as you would in any binary search tree based on comparison with existing nodes.
- Update the balance factor of each node along the path from the deleted node to the root.
The AVL tree is a powerful data structure that provides efficient search, insertion, and deletion operations while maintaining balance. Its self-balancing nature ensures that performance remains consistent even for large datasets. By incorporating proper rotation techniques, AVL trees can handle dynamic operations effectively and are widely used in various applications where sorted data is crucial.