  We give new general formulas for the asymptotics of the number of spanning
trees of a large graph. A special case answers a question of McKay (1983) for
regular graphs. The general answer involves a quantity for infinite graphs that
we call "tree entropy", which we show is a logarithm of a normalized
determinant of the graph Laplacian for infinite graphs. Tree entropy is also
expressed using random walks. We relate tree entropy to the metric entropy of
the uniform spanning forest process on quasi-transitive amenable graphs,
extending a result of Burton and Pemantle (1993).
