On the Steiner tree connecting a fractal set

We construct an example of an infinite planar embedded self-similar binary tree $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$, so that $\mathrm{\Sigma }$ is an irreducible solution to the Steiner problem with datum $C$ (i.e. $\mathrm{\Sigma }\setminus C$ is connected).