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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.
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.
The algorithms for bulk operations aren't just applicable to the red–black tree, but can be adapted to other sorted sequence data structures also, like the 2–3 tree, 2–3–4 tree and (a,b)-tree. In the following different algorithms for bulk insert will be explained, but the same algorithms can also be applied to removal and update.
Self-balancing binary trees solve this problem by performing transformations on the tree (such as tree rotations) at key insertion times, in order to keep the height proportional to log 2 (n). Although a certain overhead is involved, it is not bigger than the always necessary lookup cost and may be justified by ensuring fast execution of all ...
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.
Since AVL trees check the balance value on every insertion/deletion, it is typically stored in each node; scapegoat trees are able to calculate it only as needed, which is only when a scapegoat needs to be found. Unlike most other self-balancing search trees, scapegoat trees are entirely flexible as to their balancing.
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 ...