Generalized Outerplanar Tur\'an numbers and maximum number of k-vertex subtrees

TL;DR

This paper establishes the asymptotic maximum number of k-vertex subtrees in binary trees and relates it to the maximum number of certain cycles in outerplanar graphs, settling the generalized outerplanar Turán number for all cycles.

Contribution

It provides the first asymptotic results on the maximum number of k-vertex subtrees and cycles in outerplanar graphs, linking these to Catalan numbers.

Findings

Maximum number of k-vertex subtrees in binary trees characterized asymptotically.

Determined the maximum number of (k+2)-cycles in n-vertex outerplanar graphs.

Established the exponential growth rate of the generalized outerplanar Turán number of paths.

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 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.