Here is the detailed explanation of AVL Trees:
An AVL (Adelson-Velsky and Landis) tree is a self-balancing binary search tree.
• In an AVL tree, the heights of the two child subtrees of any node differ by at most one, ensuring that the tree remains balanced. Whenever an element is inserted or deleted, the tree is restructured to maintain its balance.
Height Balance Property:
The height-balance property is a characteristic of certain self-balancing binary trees, particularly AVL trees that ensures that the difference in height between the left and right subtrees of any node in the tree is limited to a small constant, typically -1, 0, or 1. In other words, the balance factor of each node must be within the specified range.
Balance Factor:
Each node in an AVL tree has a balance factor, which is the difference between the heights of its left and right subtrees. The balance factor should be -1, 0, or 1 to maintain balance.
Rotations:
In an AVL tree, for every node, the heights of the left and right subtrees can differ by at most 1. If this balance condition is violated after an insertion or deletion operation, rotations are performed to rebalance the tree while preserving the binary search tree property.
There are four possible rotations:
1.) Left Rotation:
This rotation is performed when the right subtree of a node becomes heavier than the left subtree. It involves making the right child of the node the new root of the subtree, moving the original root to the left child of the new root, and adjusting the subtrees accordingly.
2.) Right Rotation:
This rotation is performed when the left subtree of a node becomes heavier than the right subtree. It involves making the left child of the node the new root of the subtree, moving the original root to the right child of the new root, and adjusting the subtrees accordingly.
3.) Left-Right Rotation (Double Rotation):
This rotation is a combination of a left rotation followed by a right rotation. It’s used to balance a subtree where the left child is heavier than the right child and the left child’s right subtree is heavier than its left subtree.
4.) Right-Left Rotation (Double Rotation):
This rotation is a combination of a right rotation followed by a left rotation. It’s used to balance a subtree where the right child is heavier than the left child and the right child’s left subtree is heavier than its right subtree.
Why we use AVL Trees:
‣ Efficient Search, Insertion, and Deletion:
AVL trees provide efficient search, insertion, and deletion operations with a time complexity of O(log n), where n is the number of nodes in the tree.
‣ Balanced Structure:
The self-balancing property ensures that the tree remains balanced even after a series of insertions and deletions. This helps prevent degeneration into a linked list, which can happen in simple binary search trees.
‣ Predictable Performance:
The worst-case height of an AVL tree is guaranteed to be O(log n), providing predictable and consistent performance for various operations.
Applications of AVL Trees:
- Database Indexing
- Memory Management
- File Systems
- Game Trees
Advantages of AVL Trees:
- AVL trees can self-balance themselves.
- It is surely not skewed.
- It provides faster lookups than Red-Black Trees
- Better searching time complexity compared to other trees like binary tree.
- Height cannot exceed log(N), where, N is the total number of nodes in the tree.
Disadvantages of AVL Trees:
- It is difficult to implement.
- It has high constant factors for some of the operations.
- Less used compared to Red-Black trees.
- Due to its rather strict balance, AVL trees provide complicated insertion and removal operations as more rotations are performed.
- Take more processing for balancing.
