Generalized planar turán numbers

In a generalized Turán problem, we are given graphs H and F and seek to maximize the number of copies of H in an n-vertex graph not containing F as a subgraph. We consider generalized Turan problems where the host graph is planar. In particular, we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most 2£, for all £, £ ^ 1. We also determine the order of magnitude of the maximum number of cycles of a given length in a planar C4-free graph. An exact result is given for the maximum number of 5-cycles in a C4-free planar graph. Multiple conjectures are also introduced.