Minimal common container of tree patterns

Tree patterns represent important fragments of XPath. In this paper, we show that some classes of tree patterns exhibit such a property that, given a finite number of tree patterns P1, ..., Pn, there exists another pattern P (tree pattern or DAG-pattern) such that P1, ..., Pn, are all contained in P, and for any tree pattern Q belonging to a given class C, P1, ..., Pn, are contained in Q implies P is contained in Q.