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

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.