Psi-series method in random trees and moments of high orders

TL;DR

This paper introduces a novel psi-series method for analyzing nonlinear differential equations, deriving asymptotic expansions for probabilities related to random trees and revealing new phenomena in discrete probability and algorithm analysis.

Contribution

The paper presents a new psi-series approach for nonlinear differential equations, applied to random trees, providing explicit asymptotic expansions and uncovering novel phenomena.

Findings

Derived asymptotic expansion for the probability two random binary search trees are identical.

Showed similar asymptotics for a quantity in phylogenetic tree analysis.

Demonstrated the general applicability of the psi-series method to various problems.

Abstract

An unusual and surprising expansion of the form \[ p_n = \rho^{-n-1}(6n +\tfrac{18}5+ \tfrac{336}{3125} n^{-5}+\tfrac{1008}{3125} n^{-6} +\text{smaller order terms}), \] as $n\to\infty$, is derived for the probability $p_n$ that two randomly chosen binary search trees are identical (in shape and in labels of all corresponding nodes). A quantity arising in the analysis of phylogenetic trees is also proved to have a similar asymptotic expansion. Our method of proof is new in the literature of discrete probability and analysis of algorithms, and based on the psi-series expansions for nonlinear differential equations. Such an approach is very general and applicable to many other problems involving nonlinear differential equations; many examples are discussed and several attractive phenomena are discovered.