Statistics on non-crossing trees

Abstract

A non-crossing tree is a tree drawn on the plane having as vertices a set of points on the boundary of a circle, and whose edges are straight line segments and do not cross. Continuing previous research on non-crossing trees, we study several new statistics: number of end-points and boundary edges, number of vertices of a given degree, maximum degree, height, and path-length. In some cases, we obtain closed formulae while in others we deduce asymptotic estimates. Our approach is based on generating functions and on several bijections between NC-trees and various other combinatorial objects.