Average-case analysis of equality of binary trees under the BST probability model

Abstract

In this paper a simple algorithm to test equality of binary trees currently used in symbolic computation, unification, etc. is investigated. When the uniform probability model for the input is assumed, it is well known that it takes O(1) steps on average to decide if the trees in a given pair of total size n are equal or not. The aim of the paper is to analyze this average complexity when the so-called bst probability model is assumed. The analysis is itself more complex although feasible, involving the solution of partial differential equations and singularity analysis of Bessel functions. Nevertheless, since partial differential equations are generally unsolvable, like the one which is derived from the bivariate recurrence for the equality, an indirect mechanism solving a simpler equation and showing asymptotic equivalence of solutions is used to obtain the main result : testing equality of a pair of binary trees of total size n and distributed accordingly to the bst probability model is ⊝(logn) on the average.

This research was partially supported by the ESPRIT BRA Program of the EC under contract no. 3075, project ALCOM.