Optimum lopsided binary trees

Abstract

Binary search trees with costs α and β, respectively, on the left and right edges (lopsided search trees) are considered. The exact shape, minimum worst-case cost, and minimum average cost of lopsided trees of n internal nodes are determined for nonnegative α and β; the costs are both roughly log p ( n + 1) where p is the unique real number in the interval (1. 2] satisfying 1/ p α + 1/ p β = 1. Search procedures are given that come within a small additive constant of the lower bounds. Almost-optimum algorithms for the lopsided case of unbounded searching are also obtained. Some extensions to nonconstant costs are briefly sketched.