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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 \emph{$T_r$-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 \tcr{matching} gene tree and species tree labeled topologies. For each $T_r$ with size $r\leq 8$, 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.
trees, phylogenetics, lineage sorting, coalescent

N. A. Rosenberg, "Coalescent Histories for Caterpillar-Like Families," in IEEE/ACM Transactions on Computational Biology and Bioinformatics.
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