  In this paper, we introduce a notion of unistructural cluster algebras, for
which the set of cluster variables uniquely determines the clusters. We prove
that cluster algebras of Dynkin type and cluster algebras of rank 2 are
unistructural, then prove that if $\mathcal{A}$ is unistructural or of
Euclidean type, then $f: \mathcal{A}\to \mathcal{A}$ is a cluster automorphism
if and only if $f$ is an automorphism of the ambient field which restricts to a
permutation of the cluster variables. In order to prove this result, we also
investigate the Fomin-Zelevinsky conjecture that two cluster variables are
compatible if and only if one does not appear in the denominator of the Laurent
expansions of the other.
