  Let $1 < p < \infty$ and suppose that we are given a function $f$ defined on
the leaves of a weighted tree. We would like to extend $f$ to a function $F$
defined on the entire tree, so as to minimize the weighted $W^{1,p}$-Sobolev
norm of the extension. An easy situation is when $p = 2$, where the harmonic
extension operator provides such a function $F$. In this note we record our
analysis of the particular case of a radially-symmetric binary tree, which is a
complete, finite, binary tree with weights that depend only on the distance
from the root. Neither the averaging operator nor the harmonic extension
operator work here in general. Nevertheless, we prove the existence of a linear
extension operator whose norm is bounded by a constant depending solely on $p$.
This operator is a variant of the standard harmonic extension operator, and in
fact it is harmonic extension with respect to a certain Markov kernel
determined by $p$ and by the weights.
