A Tree Version of Konig's Theorem

König's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalisation, in which the point in one fixed side of the graph of each edge is replaced by a subtree of a given tree. The proof uses a recent extension of Hall's theorem to families of hypergraphs, by the first author and P. Haxell [3]. 1. Width and matching width of families of trees A matching in a hypergraph is a set of disjoint edges. In [3] (see also [2] and [5]) there were introduced a few notions which proved to be fruitful in the study of matchings. The two main ones are those of width and matching width, which are presented below. Let H and F be two hypergraphs on the same vertex set.