  We construct an example of an infinite planar embedded self-similar binary
tree $\Sigma$ which is the essentially unique solution to the Steiner problem
of finding the shortest connection of a given planar self-similar fractal set
$C$ of positive Hausdorff dimension. The set $C$ can be considered the set of
leaves, or the ``boundary``, of the tree $\Sigma$, so that $\Sigma$ is an
irreducible solution to the Steiner problem with datum $C$ (i.e.
$\Sigma\setminus C$ is connected).
