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The probabilities of rooted tree-shapes generated by random bifurcation

Published online by Cambridge University Press:  01 July 2016

E. F. Harding*
Affiliation:
University of Cambridge

Abstract

The set of rooted trees, generated by random bifurcation at the terminal nodes, is considered with the aims of enumerating it and of determining its probability distribution. The account of enumeration collates much previous work and attempts a complete perspective of the problems and their solutions. Asymptotic and numerical results are given, and some unsolved problems are pointed out. The problem of ascertaining the probability distribution is solved by obtaining its governing recurrence equation, and numerical results are given. The difficult problem of determining the most probable tree-shape of given size is considered, and for labelled trees a conjecture at its solution is offered. For unlabelled shapes the problem remains open. These mathematical problems arise in attempting to reconstruct evolutionary trees by the statistical approach of Cavalli-Sforza and Edwards.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1971 

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References

Brown, W. G. (1965) Historical note on a recurrent combinatorial problem. Amer. Math. Monthly 72, 973977.CrossRefGoogle Scholar
Cavalli-Sforza, L. L., Barrai, I. and Edwards, A. W. F. (1964) Analysis of human evolution under random genetic drift. Cold Spring Harbour Symposia on Quantitative Biology. XXIX, 920 CrossRefGoogle Scholar
Cavalli-Sforza, L. L. and Edwards, A. W. F. (1964) Analysis of human evolution. Genetics Today. Proc. of the XI Internat. Congress of Genetics, The Hague, The Netherlands, September 1963. Pergamon Press.Google Scholar
Cavalli-Sforza, L. L. and Edwards, A. W. F. (1966) Estimation procedures for evolutionary branching processes. Bull. Internat. Statist. Inst. (Proc. 35th session of the I.S.I.) 41, 803808.Google Scholar
Cavalli-Sforza, L. L. and Edwards, A. W. F. (1967) Phylogenetic analysis. Amer. J. Hum. Genet. 19, 233257. Also in Evolution 21, 550–570.Google ScholarPubMed
Cayley, A. (1857) On the theory of the analytic forms called trees. Phil. Mag. XIII, 172176.Google Scholar
Cayley, A. (1859) On the theory of the analytic forms called trees. Phil. Mag. XVIII, 374378.Google Scholar
Cayley, A. (1881) On the analytical forms called trees. Amer. J. Math. IV, 266268.CrossRefGoogle Scholar
Edwards, A. W. F. (1970) Likelihood estimation of the branch points of a branching diffusion process. J. R. Statist. Soc. B 32 155174.Google Scholar
Edwards, A. W. F. and Cavalli-Sforza, L. L. (1964) Reconstruction of evolutionary trees. Phenetic and Phylogenetic Analysis. Systematics Ass. Pub. No. 6, 6776.Google Scholar
Edwards, A. W. F. and Cavalli-Sforza, L. L. (1965) A method for cluster analysis. Biometrics, 21, 362375.CrossRefGoogle ScholarPubMed
Etherington, I. M. H. (1937) Non-associative powers and a functional equation. Math. Gaz. 21, 3639.CrossRefGoogle Scholar
Etherington, I. M. H. (1939) On non-associative combinations. Proc. Roy. Soc. Edinburgh 59, 153162.CrossRefGoogle Scholar
Etherington, I. M. H. (1949) Non-associative arithmetics. Proc. Roy. Soc. Edinburgh A, 62, 442453.Google Scholar
Etherington, I. M. H. (1960) Enumeration of indices of given altitude and degree. Proc. Edinburgh Math. Soc. 12, 15.CrossRefGoogle Scholar
Grossman, J. N. (1962) Bifurcating Root-trees. , University of Florida.Google Scholar
Minc, H. (1959) Enumeration of indices of given altitude and potency. Proc. Edinburgh Math. Soc. 11, 207209.CrossRefGoogle Scholar
Minc, H. (1960) Mutability of bifurcating root-trees. Quart. J. Math. 11, 187192.CrossRefGoogle Scholar
Otter, R. (1948) The number of trees. Ann. Math. 49, 583599.CrossRefGoogle Scholar
Otter, R. (1949) The multiplicative process. Ann. Math. Statist. 20, 206224.CrossRefGoogle Scholar
Pólya, G. (1937) Kombinatorische Anzahlbestimmungen für Gruppen, Graphen, und chemische Verbindungen. Acta Math. 68, 145253.CrossRefGoogle Scholar
Therianos, S. (1968) An inequality on bifurcating root-trees. Unpublished note. University of California, Santa Barbara.Google Scholar
Wedderburn, J. H. (1922) The functional equation g (x 2) = 2x + [g (x)]2 . Ann. Math. 24, 121140.CrossRefGoogle Scholar
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