Noncrossing trees and noncrossing graphs (Q2501001)

Summary: We give a parity reversing involution on noncrossing trees that leads to a combinatorial interpretation of a formula on noncrossing trees and symmetric ternary trees in answer to a problem proposed by Hough. We use the representation of Panholzer and Prodinger for noncrossing trees and find a correspondence between a class of noncrossing trees, called proper noncrossing trees, and the class of symmetric ternary trees. The second result of this paper is a parity reversing involution on connected noncrossing graphs which leads to a relation between the number of noncrossing trees with \(n\) edges and \(k\) descents and the number of connected noncrossing graphs with \(n+1\) vertices and \(m\) edges.