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To split an AVL tree into two smaller trees, those smaller than key k, and those greater than key k, first draw a path from the root by inserting k into the AVL. After this insertion, all values less than k will be found on the left of the path, and all values greater than k will be found on the right.
The weak AVL tree is defined by the weak AVL rule: Weak AVL rule: all rank differences are 1 or 2, and all leaf nodes have rank 0. Note that weak AVL tree generalizes the AVL tree by allowing for 2,2 type node. A simple proof shows that a weak AVL tree can be colored in a way that represents a red-black tree.
AVL trees and red–black trees are two examples of binary search trees that use the left rotation. A single left rotation is done in O(1) time but is often integrated within the node insertion and deletion of binary search trees. The rotations are done to keep the cost of other methods and tree height at a minimum.
The insertion and deletion algorithms, when making use of join can be independent of balancing schemes. For an insertion, the algorithm compares the key to be inserted with the key in the root, inserts it to the left/right subtree if the key is smaller/greater than the key in the root, and joins the two subtrees back with the root.
Deletion from an AVL tree may be carried out by rotating the node to be deleted down into a leaf node, and then pruning off that leaf node directly. Since at most log n nodes are rotated during the rotation into the leaf, and each AVL rotation takes constant time, the deletion process in total takes O(log n) time.
Unlike the balance information in AVL trees (using information about the height of subtrees) and red–black trees (which store a fictional "color" bit), the bookkeeping information in a WBT is an actually useful property for applications: the number of elements in a tree is equal to the size of its root, and the size information is exactly the ...
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If a large proportion of the elements of the tree are deleted, then the tree will become much larger than the current size of the stored elements, and the performance of other operations will be adversely affected by the deleted elements. When this is undesirable, the following algorithm can be followed to remove a value from the 2–3–4 tree: