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Isomorphism and Symmetries in Random Phylogenetic Trees

Part of: Combinatorial probability Combinatorics

Published online by Cambridge University Press:  14 July 2016

Miklós Bóna  and
Philippe Flajolet
Miklós Bóna*
Affiliation:
University of Florida
Philippe Flajolet*
Affiliation:
INRIA
*
∗Postal address: Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville, FL 32611–8105, USA.
∗∗Postal address: Algorithms Project, INRIA Rocquencourt, F-78153 Le Chesnay, France. Email address: philippe.flajolet@inria.fr
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Abstract

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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.

Keywords

Analytic combinatoricscombinatorial probabilitygenerating functionisomorphism problemlimit distributionsnonplane treephylogeniesrandom treesingularity analysissymmetries and automorphisms

MSC classification

Secondary: 60C05: Combinatorial probability 05A16: Asymptotic enumeration 05A15: Exact enumeration problems, generating functions
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 1005 - 1019
Copyright
Copyright © Applied Probability Trust 2009 

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