A set of ancestral configurations can be associated with a given gene tree topology $G$ and species tree topology $S$ with bijectively labeled leaves from the fixed set $X$. At a specific node in the seed tree. We introduce lattice structures into ancestral constructions by studying directed graphs, which provide a graphical representation of lattices of ancestral constructions. For congruent gene-tree and species-tree topologies, we present a method to define a directed graph of ancestral configurations from tree topologies using iterative Cartesian product of graphs. We show that a given set of paths on a directed graph of ancestral composition is bijective with a set of labeled histories. This is a well-known phylogenetic object that enumerates the possible temporal order of tree coalescence. For each set of tree families, we obtain a closed-form expression for the number of labeled histories by counting the associated directed graph paths using this bijection. Finally, we prove that our lattice structure extends to discordant tree pairs and use it to feature the pair $(G,S)$ with the largest number of ancestral configurations for a fixed $G$ Is included. We describe how construction provides a new method for performing enumeration of combinatorial aspects of gene trees and species trees.
