  To a given gene tree topology $G$ and species tree topology $S$ with leaves
labeled bijectively from a fixed set $X$, one can associate a set of ancestral
configurations, each of which encodes a set of gene lineages that can be found
at a given node of the species tree. We introduce a lattice structure on
ancestral configurations, studying the directed graphs that provide graphical
representations of lattices of ancestral configurations. For a matching gene
tree topology and species tree topology $G=S$, we present a method for defining
the digraph of ancestral configurations from the tree topology by using
iterated cartesian products of graphs. We show that a specific set of paths on
the digraph of ancestral configurations is in bijection with the set of labeled
histories -- a well-known phylogenetic object that enumerates possible temporal
orderings of the coalescences of a tree. For each of a series of tree families,
we obtain closed-form expressions for the number of labeled histories by using
this bijection to count paths on associated digraphs. Finally, we prove that
our lattice construction extends to nonmatching tree pairs, and we use it to
characterize pairs $(G,S)$ having the maximal number of ancestral
configurations for a fixed $G$. We discuss how the construction provides new
methods for performing enumerations of combinatorial aspects of gene and
species trees.
