  A strong edge coloring of a graph $G$ is a proper edge coloring of $G$ such
that every color class is an induced matching. The minimum number of colors
required is termed the strong chromatic index. In this paper, we determine the
exact value of the strong chromatic index of all unitary Cayley graphs. Our
investigations reveal an underlying product structure from which the unitary
Cayley graphs emerge. We then go on to give tight bounds for the strong
chromatic index of the Cartesian product of two trees, including an exact
formula for the product in the case of stars. Further, we give bounds for the
strong chromatic index of the product of a tree with a cycle. For any tree,
those bounds may differ from the actual value only by not more than a small
additive constant (at most 2 for even cycles and at most 5 for odd cycles),
moreover they yield the exact value when the length of the cycle is divisible
by $4$.
