Holonomic equations and efficient random generation of binary trees

TL;DR

This paper extends the use of holonomic equations for efficient random generation from binary trees to Motzkin and Schr{"o}der trees, achieving linear expected complexity under certain conditions.

Contribution

It introduces algorithms for random generation of Motzkin and Schr{"o}der trees based on holonomic equations, with proven linear or near-linear expected complexity.

Findings

Schr{"o}der tree generator has linear expected complexity.

Motzkin tree generator achieves O(n) complexity with an efficient oracle.

Proposed algorithms are practical for large tree sizes up to ten million.

Abstract

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