Generalised outerplanar Turán numbers and maximum number of k-vertex subtrees

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 generalised outerplanar Turan number for all cycles. We also determine the exponential growth of the generalised outerplanar Turan number of paths P k as a function of k which implies the order of magnitude of the generalised outerplanar Turan number of arbitrary trees. The bounds are strongly related to the sequence of Catalan numbers.