Annals of Combinatorics

, Volume 10, Issue 1, pp 129–146

The Mean and Variance of the Numbers of r-Pronged Nodes and r-Caterpillars in Yule-Generated Genealogical Trees

Original Paper

DOI: 10.1007/s00026-006-0278-6

Cite this article as:
Rosenberg, N.A. Ann. Comb. (2006) 10: 129. doi:10.1007/s00026-006-0278-6

Abstract.

The Yule model is a frequently-used evolutionary model that can be utilized to generate random genealogical trees. Under this model, using a backwards counting method differing from the approach previously employed by Heard (Evolution 46: 1818–1826), for a genealogical tree of n lineages, the mean number of nodes with exactly r descendants is computed (2 ≤  r ≤  n − 1). The variance of the number of r-pronged nodes is also obtained, as are the mean and variance of the number of r-caterpillars. These results generalize computations of McKenzie and Steel for the case of r = 2 (Math. Biosci. 164: 81–92, 2000). For a given n, the two means are largest at r = 2, equaling 2n/3 for n ≥  5. However, for n ≥  9, the variances are largest at r = 3, equaling 23n/420 for n ≥  7. As n→∞, the fraction of internal nodes that are r-caterpillars for some r approaches (e2 − 5)/4≈ 0.59726.

Keywords.

binary search treecherriescoalescentgenealogylabeled topologypectinate

AMS Subject Classification.

05C0592D15

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of Human Genetics and Bioinformatics ProgramUniversity of MichiganAnn ArborUSA