  For a fixed set $X$ containing $n$ taxon labels, an ordered pair consisting
of a gene tree topology $G$ and a species tree $S$ bijectively labeled with the
labels of $X$ possesses a set of coalescent histories -- mappings from the set
of internal nodes of $G$ to the set of edges of $S$ describing possible lists
of edges in $S$ on which the coalescences in $G$ take place. Enumerations of
coalescent histories for gene trees and species trees have produced suggestive
results regarding the pairs $(G,S)$ that, for a fixed $n$, have the largest
number of coalescent histories. We define a class of 2-cherry binary tree
topologies that we term $p$-pseudocaterpillars, examining coalescent histories
for non-matching pairs $(G,S)$, in the case in which $S$ has a caterpillar
shape and $G$ has a $p$-pseudocaterpillar shape. Using a construction that
associates coalescent histories for $(G,S)$ with a class of "roadblocked"
monotonic paths, we identify the $p$-pseudocaterpillar labeled gene tree
topology that, for a fixed caterpillar labeled species tree topology, gives
rise to the largest number of coalescent histories. The shape that maximizes
the number of coalescent histories places the "second" cherry of the
$p$-pseudocaterpillar equidistantly from the root of the "first" cherry and
from the tree root. A symmetry in the numbers of coalescent histories for
$p$-pseudocaterpillar gene trees and caterpillar species trees is seen to exist
around the maximizing value of the parameter $p$. The results provide insight
into the factors that influence the number of coalescent histories possible for
a given gene tree and species tree.
