Acyclic database schemes have attracted much interest because of the nice properties enjoyed by such schemes. Recently some new acyclicity conditions that are strictly stronger than the normal $\alpha$-acyclicity have been introduced by Fagin. Because of increased requirements, the database schemes in the new classes have some further useful properties that are not shared by $\alpha$-acyclic schemes. Therefore the new classes have practical relevance. A database designer may work in terms of attribute sets and data dependencies, and not only in terms of database schemes. Thus it is important to have a characterization for the acyclic schemes of various degree in terms of data dependencies. For $\alpha$-acyclic schemes such a characterization exists, but for the new classes the question has been open. In this paper we provide characterizations for $\beta$-, $\gamma$- and Berge-acyclic database schemes. The characterizations can be stated in a simple form: thus they should be useful for the database designer.