Isomorphism and Symmetries in Random Phylogenetic Trees

Abstract

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.

Psi-series method for equality of random trees and quadratic convolution recurrences

“…Recently, there has been also an increasing interest in statistics defined on two random combinatorial objects; see Ref. 7 and the references therein.…”

“…1. Left: the BST constructed from the sequence [6,2,4,8,7,1,5,3,10,9]. Right: the root assumes the value j with equal probability 1/n for j = 1, .…”

“…As far as we were aware, the asymptotic expansion (7) with missing terms is rare in the analysis of algorithms and applied probability literature. The expansion also indicates that the approximation of p n ρ n+1 by the first two terms 6n + 18 5 is numerically very precise as can be seen in Fig.…”

“…In addition to the unusual form of (7) and its theoretical value per se, the interest of such a psiseries expansion is multifold. First, since no analytic form for the movable singularity ρ is available, the psi-series expansion provides an effective means of obtaining an approximate numerical value to ρ; see (25) on page 85 for more details.…”

The distribution of height and diameter in random non‐plane binary trees

“…Such trees cannot be generated by a branching process conditioned by size and no direct random walk approach appears to be possible, due to the inherent presence of symmetries. (An analysis of such symmetries otherwise occurs in the recent article [6]. ) The analysis of unlabelled non‐plane trees finds its origins in the works of Pólya [36] and Otter [35].…”