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Most operations on a binary search tree (BST) take time directly proportional to the height of the tree, so it is desirable to keep the height small. A binary tree with height h can contain at most 2 0 +2 1 +···+2 h = 2 h+1 −1 nodes. It follows that for any tree with n nodes and height h: + And that implies:
Various height-balanced binary search trees were introduced to confine the tree height, such as AVL trees, Treaps, and red–black trees. [5] The AVL tree was invented by Georgy Adelson-Velsky and Evgenii Landis in 1962 for the efficient organization of information. [6] [7] It was the first self-balancing binary search tree to be invented. [8]
The preliminary steps for deleting a node are described in section Binary search tree#Deletion. There, the effective deletion of the subject node or the replacement node decreases the height of the corresponding child tree either from 1 to 0 or from 2 to 1, if that node had a child.
A weight-balanced tree is a binary search tree that stores the sizes of subtrees in the nodes. That is, a node has fields key, of any ordered type; value (optional, only for mappings) left, right, pointer to node; size, of type integer. By definition, the size of a leaf (typically represented by a nil pointer) is zero.
In computer science, an optimal binary search tree (Optimal BST), sometimes called a weight-balanced binary tree, [1] is a binary search tree which provides the smallest possible search time (or expected search time) for a given sequence of accesses (or access probabilities). Optimal BSTs are generally divided into two types: static and dynamic.
English: Analysis of data structures, tree compared to hash and array based structures, height balanced tree compared to more perfectly balanced trees, a simple height balanced tree class with test code, comparable statistics for tree performance, statistics of worst case strictly-AVL-balanced trees versus perfect full binary trees.
A binary search tree is said to be weight-balanced if half the nodes are on the left of the root, and half on the right. An α-weight-balanced node is defined as meeting a relaxed weight balance criterion:
An AVL tree is a kind of balanced binary search tree in which the two children of each internal node must have heights that differ by at most one. [7] The height of an external node is zero, and the height of any internal node is always one plus the maximum of the heights of its two children.