 Solution

Exhaust ${\mathbb{Z}}^2$ by finite boxes $G_n$ and let $T_n$ be a uniform spanning tree of $G_n$. Every finite tree satisfies the handshaking lemma, soChoose the root uniformly from $V(G_n)$. The proportion of roots within any fixed distance of the boundary tends to zero, and the rooted trees converge locally to the uniform spanning tree $T$ of ${\mathbb{Z}}^2$. Since every degree is at most four, expectations also converge. Translation invariance therefore givesThe four edges incident to $0$ have equal inclusion probability by the rotations and reflections of the square lattice. If that common probability is $r$, then $4r=\mathbb{E}{\mathrm{deg}}_T(0)=2$. Consequentlyfor every edge $e$ by translation invariance.