The Sobolev extension problem on trees and in the plane

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 $\mathrm{\Phi }$ on all of $V$ so that $\mathrm{\mid }\mathrm{\mid }\mathrm{\Phi }\mathrm{\mid }{\mathrm{\mid }}_{L^{1,p}(V)}$ has optimal order of magnitude. Here, $L^{1,p}(V)$ is a weighted Sobolev space on $V$. (2) Given a function $f:E\to \mathbb{R}$, extend it to a function $F\in L^{2,p}({\mathbb{R}}^2)$ so that $\mathrm{\mid }\mathrm{\mid }F\mathrm{\mid }{\mathrm{\mid }}_{L^{2,p}({\mathbb{R}}^2)}$ has optimal order of magnitude.