  Coalescent histories provide lists of species tree branches on which gene
tree coalescences can take place, and their enumerative properties assist in
understanding the computational complexity of calculations central in the study
of gene trees and species trees. Here, we solve an enumerative problem left
open by Rosenberg (IEEE/ACM Transactions on Computational Biology and
Bioinformatics 10: 1253-1262, 2013) concerning the number of coalescent
histories for gene trees and species trees with a matching labeled topology
that belongs to a generic caterpillar-like family. By bringing a generating
function approach to the study of coalescent histories, we prove that for any
caterpillar-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 the
family 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 claim
demonstrated only for seed trees with at most 8 taxa to arbitrary seed trees,
expanding the set of cases for which detailed enumerative properties of
coalescent histories can be determined. We introduce a procedure that computes
from $t$ the constant $\beta_t$ as well as the algebraic expression for the
generating function of the sequence $(h_n)_{n\geq 0}$.
