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Holonomic equations and efficient random generation of binary trees

Pierre Lescanne

TL;DR

The paper extends Rémy’s holonomic-recurrence framework for linear-time random generation from binary trees to Motzkin and Schröder trees. It develops a bijective, constructive approach based on the Motzkin (Dulucq–Penaud) and Foata–Zeilberger proofs, introducing leaf- and node-marked slanting-tree structures and concrete vector-based algorithms. A constant-time oracle (or a practical approximation) enables linear-time generation for Motzkin trees on practical sizes, while Schröder-tree generation relies on a related bijection to achieve quasi-linear complexity with a controlled retry mechanism. The work provides implementations in Haskell and Python, benchmarks up to about $9 imes10^6$–$10^7$ size, and situates the results within the broader landscape of Boltzmann samplers and linear-time random-generation methods. Overall, it demonstrates that holonomic recurrences can guide efficient, exact-size random generation across several tree families, with concrete practical performance benefits.

Abstract

Holonomic equations are recursive equations which allow computing efficiently numbers of combinatoric objects. R{é}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{ö}der trees and show that despite slight differences my algorithm that generates random Schr{ö}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.

Holonomic equations and efficient random generation of binary trees

TL;DR

The paper extends Rémy’s holonomic-recurrence framework for linear-time random generation from binary trees to Motzkin and Schröder trees. It develops a bijective, constructive approach based on the Motzkin (Dulucq–Penaud) and Foata–Zeilberger proofs, introducing leaf- and node-marked slanting-tree structures and concrete vector-based algorithms. A constant-time oracle (or a practical approximation) enables linear-time generation for Motzkin trees on practical sizes, while Schröder-tree generation relies on a related bijection to achieve quasi-linear complexity with a controlled retry mechanism. The work provides implementations in Haskell and Python, benchmarks up to about – size, and situates the results within the broader landscape of Boltzmann samplers and linear-time random-generation methods. Overall, it demonstrates that holonomic recurrences can guide efficient, exact-size random generation across several tree families, with concrete practical performance benefits.

Abstract

Holonomic equations are recursive equations which allow computing efficiently numbers of combinatoric objects. R{é}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{ö}der trees and show that despite slight differences my algorithm that generates random Schr{ö}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.
Paper Structure (18 sections, 30 equations, 20 figures)

This paper contains 18 sections, 30 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Random binary trees
  3. Motzkin trees
  4. Motzkin numbers and Motzkin trees
  5. Dulucq-Penaud bijection proof
  6. Slanting binary trees
  7. A taxonomy of slanting trees
  8. The bijection
  9. A concrete algorithm for random generation of Motzkin trees
  10. Linearity and oracle
  11. Schröder trees
  12. Definition of Schröder trees
  13. Foata and Zeilberger bijection
  14. A concrete algorithm for random generation of Schröder trees
  15. Ratio of failures and average linear complexity
  16. ...and 3 more sections

Figures (20)

  • Figure 1: Rémy's right insertion of a leaf
  • Figure 2: Haskell program for Rémy's algorithm
  • Figure 3: The 9 non-intersecting chords between 4 points on a circle
  • Figure 4: The 9 parenthesized words with constant c.
  • Figure 5: The 9 routes in the upper right quadrant from $(0,0)$ to $(0,4)$ with move up, down and straight
  • ...and 15 more figures