  Kirchhoff's Matrix-Tree Theorem asserts that the number of spanning trees in
a finite graph can be computed from the determinant of any of its reduced
Laplacian matrices. In many cases, even for well-studied families of graphs,
this can be computationally or algebraically taxing. We show how two well-known
results from linear algebra, the Matrix Determinant Lemma and the Schur
complement, can be used to elegantly count the spanning trees in several
significant families of graphs.
