  Let $G$ be a connected graph in which almost all vertices have linear degrees
and let $T$ be a uniform spanning tree of $G$. For any fixed rooted tree $F$ of
height $r$ we compute the asymptotic density of vertices $v$ for which the
$r$-ball around $v$ in $T$ is isomorphic to $F$. We deduce from this that if
$\{G_n\}$ is a sequence of such graphs converging to a graphon $W$, then the
uniform spanning tree of $G_n$ locally converges to a multi-type branching
process defined in terms of $W$.
  As an application, we prove that in a graph with linear minimum degree, with
high probability, the density of leaves in a uniform spanning tree is at least
$1/e-o(1)$, the density of vertices of degree $2$ is at most $1/e+o(1)$ and the
density of vertices of degree $k\geq 3$ is at most ${(k-2)^{k-2} \over (k-1)!
e^{k-2}} + o(1)$. These bounds are sharp.
