  We consider $\omega^n$-automatic structures which are relational structures
whose domain and relations are accepted by automata reading ordinal words of
length $\omega^n$ for some integer $n\geq 1$. We show that all these structures
are $\omega$-tree-automatic structures presentable by Muller or Rabin tree
automata. We prove that the isomorphism relation for $\omega^2$-automatic
(resp. $\omega^n$-automatic for $n>2$) boolean algebras (respectively, partial
orders, rings, commutative rings, non commutative rings, non commutative
groups) is not determined by the axiomatic system ZFC. We infer from the proof
of the above result that the isomorphism problem for $\omega^n$-automatic
boolean algebras, $n > 1$, (respectively, rings, commutative rings, non
commutative rings, non commutative groups) is neither a $\Sigma_2^1$-set nor a
$\Pi_2^1$-set. We obtain that there exist infinitely many $\omega^n$-automatic,
hence also $\omega$-tree-automatic, atomless boolean algebras $B_n$, $n\geq 1$,
which are pairwise isomorphic under the continuum hypothesis CH and pairwise
non isomorphic under an alternate axiom AT, strengthening a result of [FT10].
