The maximum number of paths of length four in a planar graph

Let f(n,H) denote the maximum number of copies of H in an n-vertex planar graph. The order of magnitude of f(n,Pk), where Pk is a path on k vertices, is n⌊k−12⌋+1. In this paper we determine the asymptotic value of f(n,P5) and give conjectures for longer paths.