What is the difference between binary search trees and AVL trees?

Binary search trees (BSTs) and AVL trees are both types of binary trees used for efficient searching and insertion of elements. However, there are some key differences between them.

1. Balance Factor:
- In a BST, there is no restriction on the balance of the tree. It can become highly imbalanced, leading to inefficient search and insertion operations.
- In an AVL tree, the balance factor of each node is maintained. The balance factor is the difference between the heights of the left and right subtrees of a node. It ensures that the tree remains balanced, with a maximum balance factor of 1.

2. Height-Balanced Property:
- A BST does not guarantee a balanced structure. The height of a BST can be as bad as O(n), where n is the number of elements in the tree. This worst-case scenario occurs when the tree is skewed.
- An AVL tree guarantees a balanced structure. The height of an AVL tree is always O(log n), where n is the number of elements in the tree. This ensures efficient search and insertion operations.

3. Rotations:
- In a BST, there are no rotations performed to maintain balance. The tree structure is solely determined by the order of insertion.
- In an AVL tree, rotations are performed to maintain balance whenever necessary. There are four types of rotations: left-rotation, right-rotation, left-right rotation, and right-left rotation. These rotations ensure that the balance factor of each node remains within the acceptable range.

4. Insertion and Deletion:
- In a BST, insertion and deletion operations are straightforward. The element is inserted or deleted based on its value, without considering the balance of the tree.
- In an AVL tree, insertion and deletion operations are more complex. After the element is inserted or deleted, the balance factor of each node is checked, and rotations are performed if necessary to maintain balance.

5. Performance:
- In a BST, the average case time complexity for search, insertion, and deletion operations is O(log n), where n is the number of elements in the tree. However, in the worst case, the time complexity can be O(n) if the tree is highly imbalanced.
- In an AVL tree, the time complexity for search, insertion, and deletion operations is always O(log n), guaranteeing efficient performance in all cases.

In summary, the main difference between binary search trees and AVL trees lies in their balance properties. AVL trees ensure a balanced structure through the use of balance factors and rotations, resulting in efficient search and insertion operations with a guaranteed worst-case time complexity of O(log n). On the other hand, binary search trees do not guarantee balance, leading to potential inefficiencies in certain scenarios.