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Issue No.05 - Sept.-Oct. (2013 vol.10)
pp: 1253-1262
Published by the IEEE Computer Society
Noah A. Rosenberg , Stanford University, Stanford
A coalescent history is an assignment of branches of a gene tree to branches of a species tree on which coalescences in the gene tree occur. The number of coalescent histories for a pair consisting of a labeled gene tree topology and a labeled species tree topology is important in gene tree probability computations, and more generally, in studying evolutionary possibilities for gene trees on species trees. Defining the $(T_r)$-caterpillar-like family as a sequence of $(n)$-taxon trees constructed by replacing the $(r)$-taxon subtree of $(n)$-taxon caterpillars by a specific $(r)$-taxon labeled topology $(T_r)$, we examine the number of coalescent histories for caterpillar-like families with matching gene tree and species tree labeled topologies. For each $(T_r)$ with size $(r\le 8)$, we compute the number of coalescent histories for $(n)$-taxon trees in the $(T_r)$-caterpillar-like family. Next, as $(n\rightarrow \infty)$, we find that the limiting ratio of the numbers of coalescent histories for the $(T_r)$ family and caterpillars themselves is correlated with the number of labeled histories for $(T_r)$. The results support a view that large numbers of coalescent histories occur when a tree has both a relatively balanced subtree and a high tree depth, contributing to deeper understanding of the combinatorics of gene trees and species trees.

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