  A graph $H^{\prime}$ is $(H, G)$-saturated if it is $G$-free and the addition
of any edge of $H$ not in $H^{\prime}$ creates a copy of $G$. The saturation
number $sat(H, G)$ is the minimum number of edges in a $(H, G)$-saturated
graph. We investigate bounds on the saturation number of trees $T$ in the
$n$-dimensional hypercube $Q_n$. We first present a general lower bound on the
saturation number based on the minimum degree of non-leaves. From there, we
suggest two general methods for constructing $T$-saturated subgraphs of $Q_n$,
and prove nontrivial upper bounds for specific types of trees, including paths,
generalized stars, and certain caterpillars under a restriction on minimum
degree with respect to diameter.
