Simple random sampling of binary forests with fixed number of nodes and trees

We generalize the classical algorithm of Rémy for random sampling of full binary trees with a given number of leaves. As a result, we give a simple linear time algorithm for random generation of full binary forests with given number of trees and leaves. The algorithm is obtained from a bijection that we construct in order to give a direct proof of the well-known fact that these forests are enumerated by the k-th fold self-convolution of the Catalan numbers. Via some well-known bijections, the given algorithm can be used to sample random objects from several other classes enumerated by self-convolutions of the Catalan numbers, e.g., binary forests with a given number of trees, lists of a given number of balanced strings and others.