AVLs

AVLs, also known as AVL trees, are self-balancing binary search trees. They are named after their inventors, Georg AVL and Donald Knuth. The defining characteristic of an AVL tree is that for every node, the height difference between its left and right subtrees, known as the balance factor, is at most one. This balance factor is calculated as height(left subtree) - height(right subtree). If this condition is violated at any node during an insertion or deletion operation, the tree performs rotations to restore the balance.

The primary advantage of AVL trees is their guaranteed logarithmic time complexity for search, insertion, and deletion operations, which is O(log n) where n is the number of nodes. This is because the self-balancing mechanism prevents the tree from degenerating into a linked list structure, which would result in O(n) complexity. Common operations like searching for a specific value are efficient.

Rotations are the key to maintaining the AVL property. There are four types of rotations: single left rotation, single right rotation, double left-right rotation, and double right-left rotation. These rotations rearrange the nodes in the tree to reduce the height difference between subtrees when an imbalance occurs. AVL trees are more complex to implement than standard binary search trees due to the overhead of maintaining balance. However, for applications where performance predictability and efficiency for all operations are critical, AVL trees offer a robust solution.