Generalised outerplanar Turán numbers and maximum number of [formula omitted]-vertex subtrees

Abstract

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