Coalescent Histories for Caterpillar-Like Families

摘要

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 $(rle 8)$, we compute the number of coalescent histories for $(n)$-taxon trees in the $(T_r)$-caterpillar-like family. Next, as $(nrightarrow infty)$, 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.