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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.
A Binary Search Tree is a node-based data structure where each node contains a key and two subtrees, the left and right. For all nodes, the left subtree's key must be less than the node's key, and the right subtree's key must be greater than the node's key.
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.
Fig. 1: A binary search tree of size 9 and depth 3, with 8 at the root. In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree.
The cost of a search is modeled by assuming that the search tree algorithm has a single pointer into a binary search tree, which at the start of each search points to the root of the tree. The algorithm may then perform any sequence of the following operations: Move the pointer to its left child. Move the pointer to its right child.
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:
In the theory of optimal binary search trees, the interleave lower bound is a lower bound on the number of operations required by a Binary Search Tree (BST) to execute a given sequence of accesses. Several variants of this lower bound have been proven. [1] [2] [3] This article is based on a variation of the first Wilber's bound. [4]
Optimal binary search tree; R. Random binary tree; Red–black tree; Rope (data structure) Rotation distance; S. Scapegoat tree; Segment tree; Self-balancing binary ...