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Growing binary trees

Olivier Bodini, Antoine Genitrini, Khaydar Nurligareev

Abstract

This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled structures and the analysis of polynomial iterates, we extend previous models of increasing trees with label repetitions by allowing growth branches to terminate. This mechanism enables a direct connection between dynamic evolutionary processes and classical unlabeled binary trees. We provide a combinatorial outlook for this model, linking our new approach to essential but traditionally complex parameters such as tree height, the maximum number of leaves at the deepest level (for a given tree size), and the overall tree profile. Our approach reveals structural links with Mandelbrot polynomials and coding theory. Furthermore, we leverage these structural insights to develop an efficient, iterative uniform random sampler for binary trees with a prescribed profile, achieving optimal complexity in both time and space and in random bit consumption.

Growing binary trees

Abstract

This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled structures and the analysis of polynomial iterates, we extend previous models of increasing trees with label repetitions by allowing growth branches to terminate. This mechanism enables a direct connection between dynamic evolutionary processes and classical unlabeled binary trees. We provide a combinatorial outlook for this model, linking our new approach to essential but traditionally complex parameters such as tree height, the maximum number of leaves at the deepest level (for a given tree size), and the overall tree profile. Our approach reveals structural links with Mandelbrot polynomials and coding theory. Furthermore, we leverage these structural insights to develop an efficient, iterative uniform random sampler for binary trees with a prescribed profile, achieving optimal complexity in both time and space and in random bit consumption.

Paper Structure

This paper contains 15 sections, 19 theorems, 78 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Combinatorial modeling of a growth process
  3. Growth process
  4. The set of growing binary trees
  5. Cumulative anchor counting
  6. Maximal number of anchors for a given tree size
  7. Growing binary trees of fixed height
  8. Random sampling
  9. Combinatorial context
  10. Uniform random sampling of trees with a given profile
  11. Perspectives
  12. Acknowledgments
  13. Appendix related to Section 3
  14. Appendix related to Section 4
  15. Appendix related to Section 5

Key Result

Lemma 3.1

The sequence $(a_n)_{n\geqslant1}$ satisfies the following recurrent relation:

Figures (7)

  • Figure 1: a) A growing binary tree of height $h=5$ with $m=8$ anchors. b) A growing binary tree with an additional internal node attached to its root.
  • Figure 2: Numbers $t_{n,2k,h}$ of growing binary trees of height $h=4$.
  • Figure 3: The image of a cell $(n,k)$ while passing from $S_h$ to $S_{h+1}$.
  • Figure 4: Decomposition of $\Lambda_{h+1}$ into three parts for $h=4$.
  • Figure 5: Rearranging the nonzero domain $S_h$ in the case $h=5$.
  • ...and 2 more figures

Theorems & Definitions (41)

  • Remark 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Corollary 3.3
  • Lemma 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Corollary 4.4
  • Proposition 4.5
  • ...and 31 more