A classical conjecture of Erd\H{o}s and S\'os asks to determine the Tur\'an
number of a tree. We consider variants of this problem in the settings of
hypergraphs and multi-hypergraphs. In particular, we determine the Tur\'an
number of a hypergraph without a Berge copy of a tree with k edges, for all
k and r, r≥k(k−2) and infinitely many n. We also characterize the
extremal hypergraphs for these values