Exact-size Sampling for Motzkin Trees in Linear Time via Boltzmann Samplers and Holonomic Specification

Boltzmann samplers are a kind of random samplers; in 2004, Duchon, Flajolet, Louchard and Schaeffer showed that given a combinatorial class and a combinatorial specification for that class, one can automatically build a Boltzmann sampler. In this paper, we introduce a Boltzmann sampler for Motzkin trees built from a holonomic specification, that is, a specification that uses the pointing operator. This sampler is inspired by Remy's algorithm on binary trees. We show that our algorithm gives an exact size sampler with a linear time and space complexity in average.