Isomorphism and Symmetries in Random Phylogenetic Trees

TL;DR

This paper investigates the asymptotic properties of isomorphism and symmetry in large random phylogenetic trees, revealing probabilistic laws and distributions through advanced combinatorial methods.

Contribution

It provides the first detailed asymptotic analysis of isomorphism probability and symmetry distribution in random phylogenetic trees using analytic combinatorics techniques.

Findings

Probability of isomorphism decreases exponentially with size.

Number of symmetrical nodes follows a Gaussian distribution.

Same number of symmetries obeys an inverse square-root law.

Abstract

The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.