Holonomic equations and efficient random generation of binary trees

Holonomic equations are recursive equations which allow computingefficiently numbers of combinatoric objects. R{\'e}my showed that theholonomic equation associated with binary trees yields an efficientlinear random generator of binary trees. I extend this paradigm toMotzkin trees and Schr{\"o}der trees and show that despite slightdifferences my algorithm that generates random Schr{\"o}der trees has linearexpected complexity and my algorithm that generates Motzkin trees is inO(n) expected complexity, only if we can implement a specific oraclewith a O(1) complexity. For Motzkin trees, I propose a solution whichworks well for realistic values (up to size ten millions) and yields anefficient algorithm.