Where do (random) trees grow leaves?

TL;DR

The paper introduces and analyzes the leaf-growth measure ν_{τ_n} on leaves of a uniform plane binary tree, providing an explicit construction and uncovering fractal and multifractal structures in both discrete and continuum settings. It proves a scaling limit to the Brownian CRT with a leaf measure ν_{𝒯} supported on a fractal set of Hausdorff dimension 2γ, and develops a spine-decomposition framework to study typical ν-typical leaves. A detailed discrete analysis yields the typical mass exponent γ = 3(2-√3) and a full multifractal spectrum via a β(α) function defined through an integral equation I(α,β)=0. The work also sketches a diffusion-limit program for the leaf-growth dynamics on real trees, connecting to CRT dynamics and highlighting avenues for future diffusion-approximation results. These results advance understanding of how leaf-growth procedures shape the geometry and mass distribution of large random trees and their continuous limits.

Abstract

We study a model of random binary trees grown "by the leaves" in the style of Luczak and Winkler. If $τ_n$ is a uniform plane binary tree of size $n$, Luczak and Winkler, and later explicitly Caraceni and Stauffer, constructed a measure $ν_{τ_n}$ such that the tree obtained by adding a cherry on a leaf sampled according to $ν_{τ_n}$ is still uniformly distributed on the set of all plane binary trees with size $n+1$. It turns out that the measure $ν_{τ_n}$, which we call the leaf-growth measure, is noticeably different from the uniform measure on the leaves of the tree $τ_n$. In fact, we prove that, as $n \to \infty$, with high probability it is almost entirely supported by a subset of only $n^{3 ( 2 - \sqrt{3})+o(1)} \approx n^{0.8038...}$ leaves. In the continuous setting, we construct the scaling limit of this measure, which is a probability measure on the Brownian Continuum Random Tree supported by a fractal set of dimension $ 6 (2 - \sqrt{3})$. We also compute the full (discrete) multifractal spectrum. This work is a first step towards understanding the diffusion limit of the discrete leaf-growth procedure.

Figures (5)

• Figure 1: A uniform plane binary tree $\tau_n$ with 20 000 edges decorated with its leaf-growth measure $\nu_{\tau_n}$: the black disk represents the root vertex; and the color of the other vertices (leaves) displays their $\nu_{\tau_n}$-mass (blue for small probability and red for high probability).
• Figure 2: A plane ternary tree of size 5 (i.e., with 5 internal vertices and thus $3+2\cdot 4 =11$ leaves) is grown at the leaf $l$ by adding 3 children to $l$, thus creating the tree $\operatorname{grow}(t,l)$ of size 6.
• Figure 3: A plot of the density of the discrete leaf-growth measure with respect to the discrete uniform measure on the leaves of the tree depicted in Figure \ref{['fig:intro growing tree']} (starting from the root and going counterclockwise around the tree). In the limit, those two measures are mutually singular.
• Figure 4: Illustration of the function $\alpha \to \beta( \alpha)$.
• Figure 5: A plot of the function $x \mapsto \mathsf{c}(x)$. Notice that it is symmetric with respect to $(1/2,1/2)$ and below the first bisector when $x \in [0,1/2]$.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2: Discrete multifractal spectrum
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Luczak, Winkler, 2004
• Theorem 2.4
• Remark 2.5: A sports question
• Corollary 2.6
• proof