  For every cluster-tilted algebra of simply-laced Dynkin type we provide a
companion basis which is strong, i.e. gives the set of dimension vectors of the
finitely generated indecomposable modules for the cluster-tilted algebra. This
shows in particular that every companion basis of a cluster-tilted algebra of
simply-laced Dynkin type is strong. Thus we give a proof of Parsons's
conjecture.
