which is not available in general amenable quasi-transitive graphs. The third aim of this paper is to extend these two other claims to amenable quasi-transitive graphs, that is, to prove that the formula for entropy is correct and to prove that this entropy is maximal (see \[t\].amen-ent/). We have not been able to prove uniqueness of the measure of maximal entropy. Therefore, the deduction by \[b\].BurPem/ of the uniqueness of the measure of maximal entropy for domino tilings in $\Z^2$ (their Theorem 7.2) remains incomplete. However, a new and more general proof for the uniqueness of the measure of maximal entropy for domino tilings has been found by \[b\].Sheff:thesis/. The arguments in \[b\].BurPem/ can then be used to deduce the uniqueness of the measure of maximal entropy for spanning trees in $\Z^2$ (but not in higher dimensions, which remains open).

We give general background and definitions related to graphs, groups, and entropy in \[s\].back/. In \[s\].asym/, we give our reformulation of the Matrix-Tree Theorem and its asymptotics, together with several examples. We give the formula for tree entropy as a determinant on infinite graphs in \[s\].further/, with applications. We prove the assertions on metric entropy in \[s\].ent/.

[**Note added in 