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Limits of biconditioned Bienayme-Galton-Watson trees

Vanessa Dan

Abstract

We study the limiting behavior of a Bienayme-Galton-Watson tree conditioned to have a large number of vertices and either a fixed number of leaves or a fixed number of internal nodes. The first biconditioning gives a universal result with respect to the offspring distribution. In contrast, the second case leads to a variety of limiting behaviors, ranging from condensation phenomena to more elongated tree structures, depending on the properties of the offspring distribution. To prove these results, we use tools from conditioned random walk theory and from analytic combinatorics.

Limits of biconditioned Bienayme-Galton-Watson trees

Abstract

We study the limiting behavior of a Bienayme-Galton-Watson tree conditioned to have a large number of vertices and either a fixed number of leaves or a fixed number of internal nodes. The first biconditioning gives a universal result with respect to the offspring distribution. In contrast, the second case leads to a variety of limiting behaviors, ranging from condensation phenomena to more elongated tree structures, depending on the properties of the offspring distribution. To prove these results, we use tools from conditioned random walk theory and from analytic combinatorics.

Paper Structure

This paper contains 22 sections, 27 theorems, 172 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Fixed number of leaves.
  3. Fixed number of internal nodes.
  4. Techniques.
  5. Structure of the paper.
  6. Acknowledgements.
  7. Background on rooted planar trees and BGW trees
  8. Rooted planar trees and Ł ukasiewicz paths
  9. BGW trees
  10. Large BGW trees with a fixed number of leaves
  11. Definitions
  12. Main result
  13. BGW trees with a fixed number of internal nodes
  14. Definitions
  15. Description via random walks
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\mu$ be an offspring distribution with $\mu(1)>0$ and $\mu(2)>0$. Let $k\geq 1$ and $T_n^k$ a $\mu$-BGW tree conditioned to have $n$ vertices and $k$ leaves. Then, we have the following convergence in distribution: where $R^k$ is a uniform random binary tree with $k$ leaves and $\Delta$ is a Dirichlet random variable with parameter $(1,\ldots,1)$ independent of $R^k$.

Figures (9)

  • Figure 1: An example of a tree $a$ with its reduced tree $R(a)$ and its sequence of single-child ancestors $L(a)$.
  • Figure 2: Illustration of the limit behavior of a $\mu$-BGW with $n$ nodes and $k$ leaves.
  • Figure 3: An example of a tree $a$ with its reduced tree $R(a)$ and its sequence of leaves $L(a)$.
  • Figure 4: Illustration of $D_{n,k}$ on the event $G_{n,k}$.
  • Figure 5: Illustration of the asymptotic shape of $T_{n,k}$.
  • ...and 4 more figures

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • ...and 50 more