In computer science, an AVL tree is a self-balancing binary search tree, and it is the first such data structure to be invented. In an AVL tree, the heights of the two child subtrees of any node differ by at most one; therefore, it is also said to be height-balanced. Lookup, insertion, and deletion all take O(log n) time in both the average and worst cases, where n is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more tree rotations.
The balance factor of a node is the height of its right subtree minus the height of its left subtree and a node with balance factor 1, 0, or -1 is considered balanced. A node with any other balance factor is considered unbalanced and requires rebalancing the tree. The balance factor is either stored directly at each node or computed from the heights of the subtrees.
AVL trees are often compared with red-black trees because they support the same set of operations and because red-black trees also take O(log n) time for the basic operations. AVL trees perform better than red-black trees for lookup-intensive applications. The AVL tree balancing algorithm appears in many computer science curricula.
Insertion into an AVL tree may be carried out by inserting the given value into the tree as if it were an unbalanced binary search tree, and then retracing one's steps toward the root updating the balance factor of the nodes. If the balance factor becomes -1, 0, or 1 then the tree is still in AVL form, and no rotations are necessary.
If the balance factor becomes 2 or -2 then the tree rooted at this node is unbalanced, and a tree rotation is needed. At most a single or double rotation will be needed to balance the tree.
There are basically four cases which need to be accounted for, of which two are symmetric to the other two. For simplicity, the root of the unbalanced subtree will be called P, the right child of that node will be called R, and the left child will be called L. If the balance factor of P is 2, it means that the right subtree outweighs the left subtree of the given node, and the balance factor of the right child (R) must then be checked. If the balance factor of R is 1, it means the insertion occurred on the (external) right side of that node and a left rotation is needed (tree rotation) with P as the root. If the balance factor of R is -1, this means the insertion happened on the (internal) left side of that node. This requires a double rotation. The first rotation is a right rotation with R as the root. The second is a left rotation with P as the root.
Only the nodes traversed from the insertion point to the root of the tree need be checked, and rotations are a constant time operation, and because the height is limited to O(log(n)), the execution time for an insertion is O(log(n)).
The retracing can stop if the balance factor becomes -1 or 1 indicating that the height of that subtree has remained unchanged. If the balance factor becomes 0 then the height of the subtree has decreased by one and the retracing needs to continue. If the balance factor becomes -2 or 2 then the subtree is unbalanced and needs to be rotated to fix it. If the rotation leaves the subtree's balance factor at 0 then the retracing towards the root must continue since the height of this subtree has decreased by one. This is in contrast to an insertion where a rotation resulting in a balance factor of 0 indicated that the subtree's height has remained unchanged.
The time required is O(log(n)) for lookup plus maximum O(log(n)) rotations on the way back to the root; so the operation can be completed in O(log n) time.
If each node additionally records the size of its subtree (including itself and its descendants), then the nodes can be retrieved by index in O(log n) time as well.
Once a node has been found in a balanced tree, the next or previous node can be obtained in amortized constant time. (In a few cases, about 2*log(n) links will need to be traversed. In most cases, only a single link need be traversed. On the average, about two links need to be traversed.)
The AVL tree is more rigidly balanced than Red-Black trees, leading to slower insertion and removal but faster retrieval.