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 <i>P</i>₁, ..., <i>P</i>_n, there exists another pattern <i>P</i> (tree pattern or DAG-pattern) such that <i>P</i>₁, ..., <i>P</i>_n, are all contained in <i>P</i>, and for any tree pattern <i>Q</i> belonging to a given class <i>C</i>, <i>P</i>₁, ..., <i>P</i>_n, are contained in <i>Q</i> implies <i>P</i> is contained in <i>Q</i>.