  We show that graphs generated by collapsible pushdown systems of level 2 are
tree-automatic. Even when we allow $\epsilon$-contractions and add a
reachability predicate (with regular constraints) for pairs of configurations,
the structures remain tree-automatic. Hence, their FO theories are decidable,
even when expanded by a reachability predicate. As a corollary, we obtain the
tree-automaticity of the second level of the Caucal-hierarchy.
