  One way to obtain invariants of some Legendrian submanifolds in 1-jet spaces
$J^1M$, equipped with the standard contact structure, is through the Morse
theoretic technique of generating families. This paper extends the invariant of
generating family cohomology by giving it a product $\mu_2$. To define the
product, moduli spaces of flow trees are constructed and shown to have the
structure of a smooth manifold with corners. These spaces consist of
intersecting half-infinite gradient trajectories of functions whose critical
points correspond to Reeb chords of the Legendrian. This paper lays the
foundation for an $A_\infty$ algebra which will show, in particular, that
$\mu_2$ is associative and thus gives generating family cohomology a ring
structure.
