Precise Deviations for Branches of the Random Binary Tree in the Horton–Strahler Analysis

Abstract

In this paper, we study the precise deviations for the number of branches of a random binary tree in the context of Horton–Strahler analysis. We establish precise large deviations, precise moderate deviations, and Cramér-type moderate deviations for the number of branches of the random binary tree. As a consequence of the Cramér-type moderate deviations, a Berry–Esseen bound is derived. The derivations of these results rely heavily on asymptotic analysis of certain discrete summations.