# Asymptotic Enumeration of Spanning Trees via Traces and Random Walks

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Methods of enumeration of spanning trees in a finite graph and relations to

various areas of mathematics and physics have been investigated for more

than 150 years. We will review the history and applications. Then we will

give new formulas for the asymptotics of the number of spanning trees of a

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

[Fuglede-Kadison] determinant of the graph Laplacian for infinite graphs.

Proofs involve new traces and the theory of random walks.