For a fixed set containing taxon labels, an ordered pair consisting of a gene tree topology and a species tree topology bijectively labeled with the labels of possesses a set of coalescent histories-mappings from the set of internal nodes of to the set of edges of describing possible lists of edges in on which the coalescences in take place. Enumerations of coalescent histories for gene trees and species trees have produced suggestive results regarding the pairs () that, for a fixed , have the largest number of coalescent histories. We define a class of 2-cherry binary tree topologies that we term , examining coalescent histories for non-matching pairs () in the case in which has a caterpillar shape and has a -pseudocaterpillar shape.