  Kirchhoff's matrix tree theorem is a well-known result that gives a formula
for the number of spanning trees in a finite, connected graph in terms of the
graph Laplacian matrix. A closely related result is Wilson's algorithm for
putting the uniform distribution on the set of spanning trees. We will show
that when one follows Greg Lawler's strategy for proving Wilson's algorithm,
Kirchhoff's theorem follows almost immediately after one applies some
elementary linear algebra. We also show that the same ideas can be applied to
other computations related to general Markov chains and processes on a finite
state space.
