after $k$ steps. We term $\asym(\gh)$ the [**tree entropy**]{} of $\gh$. We have chosen this terminology to reflect both its provenance as a limit of entropies per vertex of uniform spanning trees of finite graphs as well as its agreement (up to normalization) with the metric entropy of uniform spanning forests in certain infinite graphs (see \[t\].amen-ent/), as we shall discuss shortly.

Our result allows us to find asymptotics that were not known previously. For a simple example, in case the graphs $\gh_n$ approach the regular tree of degree 4, then $$\lim_{n \to\infty} |\verts(\gh_n)|^{-1}  \log \cp(\gh_n) 
=
3 \log (3/2)
\,;$$ see \[x\].wired-free/. This answers a question of \[b\].McKay/, who had shown such a result only under stronger conditions on the sequence of graphs $\gh_n$, and who had noted (in the case of regular graphs) that, if sufficient, our conditions would be weakest possible.

In the past, it has been observed by direct calculation that one has the same asymptotic complexity both of rectangular portions of $\Z^d$ as well as of $d$-dimensional tori. Our first main result, \[t\].asym/, shows that this stability phenomenon is quite general; see \[c\].subgraph/.

Our infinite series representation of the asymptotic complexity has an additional benefit. 