A cubic tree

If it could be completed to a cubic graph, it would give a cubic Moore graph of diameter 3. I wish I knew how to produce a similar tree, but growing from an edge instead of from a vertex.

\documentclass{article}
\usepackage{tikz}
\thispagestyle{empty}
\begin{document}
\newcommand{\twochildren}{child{node{\null}} child{node{\null}}}

\begin{tikzpicture}[style=thick,scale=0.5]
\tikzstyle{every node}=[circle,inner sep=2pt,draw]
\tikzstyle{level 1}=[sibling distance=20mm]
\tikzstyle{level 2}=[sibling distance=10mm]
\tikzstyle{level 3}=[sibling distance=5mm]
\node{\null}[grow'=right]
child{node{\null}
child{node{\null}\twochildren}
child{node{\null}\twochildren} }
child{node{\null}
child{node{\null}\twochildren}
child{node{\null}\twochildren} }
child{node{\null}
child{node{\null}\twochildren}
child{node{\null}\twochildren} };
\end{tikzpicture}
\end{document}