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Isomorphism and Symmetries in Random Phylogenetic Trees

Part of: Combinatorics Combinatorial probability

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: [email protected]
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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 

References

[1] Bender, E. A. (1973). Central and local limit theorems applied to asymptotic enumeration. J. Combinatorial Theory A 15, 91–111.Google Scholar
[2] Bóna, M. and Knopfmacher, A. (2008). On the probability that certain compositions have the same number of parts. To appear in Ann. Combinatorics.Google Scholar
[3] Broutin, N. and Flajolet, P. (2008). The height of random binary unlabelled trees. In 5th Colloquium Math. Comput. Sci. (Blaubeuren, 2008), ed. Rösler, U., Discrete Math. Theoret. Comput. Sci. Proc. AI, Assoc. Discrete Math. Theoret. Comput. Sci., Nancy, pp. 121–134.Google Scholar
[4] Diestel, R. (2000). Graph Theory (Graduate Texts Math. 173). Springer, New York.Google Scholar
[5] Erdős, P. and Turán, P. (1967). “On some problems of a statistical group-theory. {III}.” Acta Math. Acad. Sci. Hungar. 18, 309–320.Google Scholar
[6] Finch, S. R. (2003). Mathematical Constants. Cambridge University Press.Google Scholar
[7] Flajolet, P. and Odlyzko, A. (1982). The average height of binary trees and other simple trees. J. Comput. System Sci. 25, 171–213.Google Scholar
[8] Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press.Google Scholar
[9] Flajolet, P. et al. (2006). A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics. Electron. J. Combinatorics 13, 35pp.CrossRefGoogle Scholar
[10] Gnedenko, B. V. and Kolmogorov, A. N. (1968). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, MA.Google Scholar
[11] Goh, W. M. Y. and Schmutz, E. (1991). The expected order of a random permutation. Bull. London Math. Soc. 23, 34–42.Google Scholar
[12] Harary, F. and Palmer, E. M. (1973). Graphical Enumeration. Academic Press, New York.Google Scholar
[13] Hwang, H.-K. (1998). On convergence rates in the central limit theorems for combinatorial structures. Europ. J. Combinatorics 19, 329–343.CrossRefGoogle Scholar
[14] McKeon, K. A. (1991). The expected number of symmetries in locally-restricted trees. {I}. In Graph Theory, Combinatorics, and Applications, Vol. 2, ed. Alavi, Y., Wiley, New York, pp. 849–860.Google Scholar
[15] McKeon, K. A. (1996). The expected number of symmetries in locally restricted trees. {II}. Discrete Appl. Math. 66, 245–253.Google Scholar
[16] Nicolas, J.-L. (1985). Distribution statistique de l'ordre d'un élément du groupe symétrique. Acta Math. Hung. 45, 69–84.Google Scholar
[17] Otter, R. (1948). The number of trees. Ann. Math. 49, 583–599.Google Scholar
[18] Pólya, G. and Read, R. C. (1987). Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. Springer, New York.Google Scholar
[19] Sloane, N. J. A. (2008). The on-line encyclopedia of integer sequences. Available at www.research.att.com/∼njas/sequences/.Google Scholar
[20] Stanley, R. P. (1999). Enumerative Combinatorics, Vol. II. Cambridge University Press.Google Scholar
[21] Van Cutsem, B. and Ycart, B. (1998). Indexed dendrograms on random dissimilarities. J. Classification 15, 93–127.Google Scholar
[22] Wilf, H. S. (2005). The variance of the Stirling cycle numbers. Preprint. Available at http://arxiv.org/abs/math/0511428.Google Scholar