  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 oncrossing
trees, and the set 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 a given number of edges
and descents and the number of connected noncrossing graphs with a given number
of vertices and edges.

Combinatorics