  The notion of tree entropy was introduced by the author as a normalized limit
of the number of spanning trees in finite graphs, but is defined on random
infinite rooted graphs. We give some new expressions for tree entropy; one uses
Fuglede-Kadison determinants, while another uses effective resistance. We use
the latter to prove that tree entropy respects stochastic domination. We also
prove that tree entropy is non-negative in the unweighted case, a special case
of which establishes Lueck's Determinant Conjecture for Cayley-graph
Laplacians. We use techniques from the theory of operators affiliated to von
Neumann algebras.
