In contact topology, invariants of Legendrian submanifolds in 1-jet spaces have been obtained through a variety of techniques. I will discuss how I am enriching one Morse-theoretic invariant, Generating Family Cohomology, to an A-infinity algebra by constructing product maps. The construction uses moduli spaces of Morse flow trees: spaces of intersecting gradient trajectories of functions whose critical points encode Reeb chords of the Legendrian submanifold. I will focus my talk on the construction of a 2-to-1 product and discuss how it lays the foundation for the A-infinity algebra.