An AVL tree is a self-balancing binary search tree that ensures the height difference between its left and right subtrees is at most one. This balancing property guarantees efficient searching, insertion, and deletion operations. Let’s dive deeper into AVL trees and understand how they work.
What is a Binary Search Tree?
A binary search tree (BST) is a data structure that organizes elements in a hierarchical manner. Each element has a key value, and the left child of a node contains keys smaller than its parent, while the right child contains keys greater than its parent.
The Need for Balancing
In some cases, when elements are inserted or deleted in a BST, the tree becomes unbalanced. This imbalance can lead to skewed trees with poor performance characteristics, especially during search operations. AVL trees were introduced to address this issue by maintaining balance during insertions and deletions.
Properties of an AVL Tree
An AVL tree has the following properties:
- Height Balance: The height difference between the left and right subtrees of any node is at most one.
- Binary Search Tree Property: The left child of a node contains keys smaller than its parent, while the right child contains keys greater than its parent.
Rotations in AVL Trees
To maintain balance, AVL trees use rotations. There are four types of rotations:
- Left Rotation: This rotation is performed when the right subtree of a node becomes taller than its left subtree.
- Right Rotation: This rotation is performed when the left subtree of a node becomes taller than its right subtree.
- Left-Right Rotation: This rotation is a combination of left and right rotations, used when the left subtree of a node has a taller right subtree.
- Right-Left Rotation: This rotation is a combination of right and left rotations, used when the right subtree of a node has a taller left subtree.
Insertion in AVL Trees
When inserting an element into an AVL tree, we follow these steps:
- Perform a standard BST insertion.
- Update the heights of all ancestors starting from the newly inserted node.
- If any node violates the height balance property, perform rotations to restore balance.
Deletion in AVL Trees
When deleting an element from an AVL tree, we follow these steps:
- Perform a standard BST deletion.
- Update the heights of all ancestors starting from the deleted node’s parent.
The time complexity for search, insert, and delete operations in an AVL tree is O(log n), where n is the number of elements in the tree. This complexity remains constant even for worst-case scenarios.
AVL trees are an efficient self-balancing binary search tree data structure. Their ability to maintain balance ensures optimal performance for searching, inserting, and deleting elements.
By using rotations and keeping track of heights, AVL trees provide us with fast operations regardless of the input order.