enumeration of k-noncrossing trees and forests

a k-noncrossing tree is a noncrossing tree where each node receives a label in {1,2,…,k} such that the sum of labels along an ascent does not exceed k+1, if we consider a path from a fixed vertex called the root. in this paper, we provide a proof for a formula that counts the number of k-noncrossing trees in which the root (labelled by k) has degree d. we also find a formula for the number of forests in which each component is a k-noncrossing tree whose root is labelled by k.