On trees and noncrossing partitions

We give a simple and natural proof of (an extension of) the identity P(k, l, n) = P2(k − 1, l − 1, n − 1). The number P(k, l, n) counts noncrossing partitions of {1, 2, …,l} into n parts such that no part contains two numbers x and y, 0 < y − x