  Let $G$ be a connected $n$-vertex graph in a proper minor-closed class
$\mathcal G$. We prove that the extension complexity of the spanning tree
polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following
from the work of Wong (1980) and Martin (1991). It also extends a result of
Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound
for graphs embedded in a fixed surface. Our proof works more generally for all
graph classes admitting strongly sublinear balanced separators: We prove that
for every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph class
closed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$
have balanced separators of size $O(n^\beta)$, then the extension complexity of
the spanning tree polytope of every connected $n$-vertex graph in $\mathcal{G}$
is $O(n^{1+\beta})$. We in fact give two proofs of this result, one is a direct
construction of the extended formulation, the other is via communication
protocols. Using the latter approach we also give a short proof of the $O(n)$
bound for planar graphs due to Williams (2002).
