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Holonomic equations are recursive equations which allow computing efficientlynumbers of combinatoric objects. R{\'e}my showed that the holonomic equationassociated with binary trees yields an efficient linear random generator ofbinary trees. I extend this paradigm to Motzkin trees and Schr{\"o}der treesand show that despite slight differences my algorithm that generates randomSchr{\"o}der trees has linear expected complexity and my algorithm thatgenerates Motzkin trees is in O(n) expected complexity, only if we canimplement a specific oracle with a O(1) complexity. For Motzkin trees, Ipropose a solution which works well for realistic values (up to size tenmillions) and yields an efficient algorithm.

Holonomic equations and efficient random generation of binary trees