  In this paper, we compute asymptotics for the determinant of the
combinatorial Laplacian on a sequence of $d$-dimensional orthotope square
lattices as the number of vertices in each dimension grows at the same rate. It
is related to the number of spanning trees by the well-known matrix tree
theorem. Asymptotics for $2$ and $3$ component rooted spanning forests in these
graphs are also derived. Moreover, we express the number of spanning trees in a
$2$-dimensional square lattice in terms of the one in a $2$-dimensional
discrete torus and also in the quartered Aztec diamond. As a consequence, we
find an asymptotic expansion of the number of spanning trees in a subgraph of
$\mathbb{Z}^2$ with a triangular boundary.
