Improved Bounds on the Costs of Optimal and Balanced Binary Search Trees

Abstract

A binary search tree can be used to store data in a computer system for retrieval by name. Different elements in the tree may be referenced with different probabilities. If we define the cost of the tree as the average number of elements which must be examined in searching for an element, then different trees have different costs. We show that two particular types of trees, weight balanced trees and min-max trees, which are easily constructed from the probability distribution on the elements, are close to optimal. Specifically, we show that for any probability distribution with entropy H, H-log2H-(log2e-1)<=Copt<= Cwb ,+ H+2/Cmm,+H+2 where Copt, Cwb, and Cmm are the optimal, weigh balances, and min-max costs. We gain some added insight by deriving an expression for the expected value of the entropy of a random probability distribution.