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Coalescent histories provide lists of species tree branches on which genetree coalescences can take place, and their enumerative properties assist inunderstanding the computational complexity of calculations central in the studyof gene trees and species trees. Here, we solve an enumerative problem leftopen by Rosenberg (IEEE/ACM Transactions on Computational Biology andBioinformatics 10: 1253-1262, 2013) concerning the number of coalescenthistories for gene trees and species trees with a matching labeled topologythat belongs to a generic caterpillar-like family. By bringing a generatingfunction approach to the study of coalescent histories, we prove that for anycaterpillar-like family with seed tree $t$, the sequence $(h_n)_{n\geq 0}$describing the number of matching coalescent histories of the $n$th tree of thefamily grows asymptotically as a constant multiple of the Catalan numbers.Thus, $h_n \sim \beta_t c_n$, where the asymptotic constant $\beta_t > 0$depends on the shape of the seed tree $t$. The result extends a claimdemonstrated only for seed trees with at most 8 taxa to arbitrary seed trees,expanding the set of cases for which detailed enumerative properties ofcoalescent histories can be determined. We introduce a procedure that computesfrom $t$ the constant $\beta_t$ as well as the algebraic expression for thegenerating function of the sequence $(h_n)_{n\geq 0}$.

Asymptotic properties of the number of matching coalescent histories for caterpillar-like families of species trees