AVL Tree

The insertion and deletion operations for AVL trees begin similarly to the corresponding operations for (regular) binary search trees. It then performs post-processing structural rotations to restore the balance of any portions of the tree that are adversely affected by the insertion/removal.

Insertion

• Insert item as in binary search tree.
• Starting with new node, go up to update heights.
• If find a node that has become unbalanced, do rotation to rebalance.
• Rebalance reduces height of sub-tree by $1$, so no need to propagate up – only one adjustment needed at most because it was previously balanced before the insert.
• So, in total, $O(\lg N)$ to insert and $O(1)$ to rebalance.

Removal

• Remove item as in binary search tree.
• Starting with the parent of the deleted node, go up to update heights.
• By “deleted node,” I mean the actual deleted node (which is going to be a leaf) not the node which its value was replaced with in-order predecessor/successor.
• As you go up from the deleted node to update the heights of its ancestors, rebalance them as needed (perform rotation).
• Rotations may reduce the height of a subtree, causing further imbalances for ancestors, so you may need to perform more than one rebalancing operation.
• Since you travel from the deleted node upwards to the root, at most you will perform $O(\lg n)$ rotations.
• So, in total, $O(\lg n)$ to remove and $O(\lg n)$ to rebalance.

Recommendation: Implement insertion/removal (with structural rotations) recursively!

As you insert/remove, you must travel upward towards the root to update the height of ancestors.