Average-case Analysis of Equality of Binary Trees Under the BST Probability Model

In this paper a simple algorithm to test equality of binary trees currently used in symbolic computation, uniication, etc. is investigated. Surprisingly enough, it takes O(1) steps on average to decide if a given pair of trees of total size n are equal or not if the uniform probability model for the input is assumed. Moreover, other similar algorithms have qualitatively the same average complexity behavior. In this paper, we analyze this average complexity when the so-called bst probability model is assumed. The analysis is itself more complex although feasible, involving partial diierential equations and singularity analysis of Bessel functions. Nevertheless, partial diierential equations are generally unsolvable, like the one which is derived from the bivariate recurrences for the equality , and an indirect mechanism solving a simpler equation and showing asymptotical 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 (log n) on the average.