Associativity of a binary operation means that performing a tree rotation on it does not change the final result.

It is the poset of binary trees with n leaves, ordered by tree rotation operations.

This makes tree rotations useful for rebalancing a tree.

Insertions and deletions may require the tree to be rebalanced by one or more tree rotations.

Additionally, the tree rotations used during insertion and deletion may require updating the high value of the affected nodes.

In discrete mathematics, tree rotation is an operation on a binary tree that changes the structure without interfering with the order of the elements.

A tree rotation moves one node up in the tree and one node down.

The tree rotation renders the inorder traversal of the binary tree invariant.

If an imbalance is found, one tree rotation or pair of rotations is performed, which is guaranteed to balance the whole tree.

Tamari lattice, a partially ordered set in which the elements can be defined as binary trees and the ordering between elements is defined by tree rotation.

Self-balancing binary trees solve this problem by performing transformations on the tree (such as tree rotations) at key times, in order to keep the height proportional to log(n).

Tree rotations are used in a number of tree data structures such as AVL trees, red-black trees, splay trees, and treaps.

Balancing a k-d tree requires care because k-d trees are sorted in multiple dimensions so the tree rotation technique cannot be used to balance them as this may break the invariant.

One way to do this is to first perform a standard binary tree search for the element in question, and then use tree rotations in a specific fashion to bring the element to the top.

The information can be updated efficiently since adding a node only affects the counts of its O(log n) ancestors, and tree rotations only affect the counts of the nodes involved in the rotation.

Then, as long as x is not the root of the tree and has a larger priority number than its parent z, perform a tree rotation that reverses the parent-child relation between x and z.

There may be more than one tree for a given permutation: if two nodes that are adjacent in the same tree have the same sign, then they may be replaced by a different pair of nodes using a tree rotation operation.

Basic operations of an AVL tree involve carrying out the same actions as would be carried out on an unbalanced binary search tree, but modifications are followed by zero or more operations called tree rotations, which help to restore the height balance of the subtrees.

Rather than storing random priorities on each node, the randomized binary search tree stores at each node a small integer, the number of its descendants (counting itself as one); these numbers may be maintained during tree rotation operations at only a constant additional amount of time per rotation.