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

Determinants, their applications to Markov processes, and a random walk proof of Kirchhoff's matrix tree theorem