  We study the generalized Tur\'an function $ex(n,H,F)$, when $H$ or $F$ is
$K_{2,t}$. We determine the order of magnitude of $ex(n,H,K_{2,t})$ when $H$ is
a tree, and determine its asymptotics for a large class of trees. We also
determine the asymptotics of $ex(n,K_{2,t},H)$ in most cases.
