A database schema in the relational data model is a hypergraph whose nodes represent attributes (or column headings) and whose edges represent relations over those attributes (or tables with those column headings). Tree schemas are database schemas with a simple, treelike structure. Tree schemas are called acyclic schemes or acyclic hypergraphs in parts of the hterature. The simple structure of tree schemas is used to advantage in dwerse areas of database management, including query processing and dependency theory. This paper provides several characterizations of tree schemas. It is proved that cyclic (i.e., nontree) schemas are built from simple building blocks, called Arings and Acliques; these play a role in the theory analogous to the role of simple cycles in graph theory. It is proved that a schema is a tree schema ff and only ff it is a conformal hypergraph and a natural graph representation (the 2-section) is chordal. Indeed, conformality is equivalent to the absence of Achques, and chordality is equivalent to the absence of Axings. The present characterizations are also related to ones that appear elsewhere: acyclic hypergraphs, Graham reductions, the running intersection property, and maximal weight qual trees.
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