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