An AVL tree (Adelson-Velsky and Landis) is a self-balancing binary search tree where the height difference between the left and right subtrees of any node is at most 1. This ensures that the tree remains approximately balanced, providing efficient insertion, removal, and search operations.

AVL trees were the first self-balancing binary search trees invented, in 1962, by Georgy Adelson-Velsky and Evgenii Landis. They are widely used in applications requiring fast search and update operations, such as databases and file systems.

The main properties of an AVL tree include: automatic balancing, which keeps the tree height at O(log n), ensuring efficient operations; the ordering of elements, facilitating quick searches; and the hierarchical structure that organizes data efficiently. Rotations (single and double) are used to restore balance after insertions and removals.

Search in an AVL tree is similar to search in a standard binary search tree, leveraging the ordering of elements to quickly locate the desired value.

Insertion in an AVL tree starts like in a binary search tree, locating the correct position for the new node. After insertion, it is necessary to check the balance of the tree and, if needed, perform rotations to restore balance. To understand insertion operations in binary search trees, you can visit our binary search tree simulator.

Removal in an AVL tree also follows the process of a binary search tree, but after removal, it is necessary to check and restore the balance of the tree, if needed, using appropriate rotations.

Balance in an AVL tree is maintained by calculating the balance factor for each node, which is the difference between the heights of the left and right subtrees. If the balance factor of a node becomes greater than 1 or less than -1 after an insertion or removal, the tree is unbalanced and needs to be corrected.

There are four main cases of imbalance that can occur in an AVL tree, each requiring a specific rotation to restore balance: