Exact uniform sampling over catalan structures

We present a new framework for creating elegant algorithms for exact uniform sampling of important Catalan structures, such as triangulations of convex polygons, Dyck words, monotonic lattice paths and mountain ranges. Along with sampling, we obtain optimal coding, and optimal number of random bits required for the algorithm. The framework is based on an original two-parameter recursive relation, where Ballot and Catalan numbers appear and which may be regarded as to demonstrate a generalized reduction argument. We then describe (a) a unique $n\times n$ matrix to be used for any of the problems -the common pre-processing step of our framework- and (b) a linear height tree, where leaves correspond one by one to all distinct solutions of each problem; sampling is essentially done by selecting a path from the root to a leaf - the main algorithm. Our main algorithm is linear for a number of the problems mentioned.