  Let $V$ be a finite tree with radially decaying weights. We show that there
exists a set $E \subset \mathbb{R}^2$ for which the following two problems are
equivalent: (1) Given a (real-valued) function $\phi$ on the leaves of $V$,
extend it to a function $\Phi$ on all of $V$ so that $||\Phi||_{L^{1,p}(V)}$
has optimal order of magnitude. Here, $L^{1,p}(V)$ is a weighted Sobolev space
on $V$. (2) Given a function $f:E \rightarrow \mathbb{R}$, extend it to a
function $F \in L^{2,p}(\mathbb{R}^2)$ so that $||F||_{L^{2,p}(\mathbb{R}^2)}$
has optimal order of magnitude.
