  For a graph $F$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ (or a
Berge-$F$ in short), if there is a bijection $f : E(F) \rightarrow
E(\mathcal{H})$ such that for each $e \in E(F)$ we have $e \subset f(e)$. A
hypergraph is Berge-$F$-free if it does not contain a Berge copy of $F$. We
denote the maximum number of hyperedges in an $n$-vertex $r$-uniform
Berge-$F$-free hypergraph by $\mathrm{ex}_r(n,\textrm{Berge-}F).$
  In this paper we prove two general lemmas concerning the maximum size of a
Berge-$F$-free hypergraph and use them to establish new results and improve
several old results. In particular, we give bounds on
$\mathrm{ex}_r(n,\textrm{Berge-}F)$ when $F$ is a path (reproving a result of
Gy\H{o}ri, Katona and Lemons), a cycle (extending a result of F\"uredi and
\"Ozkahya), a theta graph (improving a result of He and Tait), or a $K_{2,t}$
(extending a result of Gerbner, Methuku and Vizer).
  We also establish new bounds when $F$ is a clique (which implies extensions
of results by Maherani and Shahsiah and by Gy\'arf\'as) and when $F$ is a
general tree.
