Ervin Győri and his coauthors initiated the systematical study of generalized Turán problems where the host graph is planar. Our motivation is to investigate a variant of these results, where the host graph is outerplanar.

We prove an asymptotic result on the maximum number of k-vertex subtrees in binary trees of given order. This problem turns out to be equivalent to determine the maximum number of k+2-cycles in n-vertex outerplanar graphs, thus we settle the generalized outerplanar Turán number for all cycles.We also determine the exponential growth of the generalized outerplanar Turán number of paths P_kas a function of k which implies the order of magnitude of the generalized outerplanar Turán number of arbitrary trees. The bounds are strongly related to the sequence of Catalan numbers.

Joint work with Dávid Matolcsi (ELTE).