We show that examinations of the expressive power of logical formulae enriched by Lindström quantifiers over ordered finite structures have a well-studied complexity-theoretic counterpart: the leaf language approach to define complexity classes. Model classes of formulae with Lindström quantifiers are nothing else than leaf language definable sets. Along the way we tighten the best up to now known leaf language characterization of the classes of the polynomial time hierarchy and give a new model-theoretic characterization of PSPACE
