Where do (random) trees grow leaves?

Abstract: We study a model of random binary trees grown "by the leaves" in the style of Luczak and Winkler. If $\tau_n$ is a uniform plane binary tree of size $n$, Luczak and Winkler, and later explicitly Caraceni and Stauffer, constructed a measure $\nu_{\tau_n}$ such that the tree obtained by adding a cherry on a leaf sampled according to $\nu_{\tau_n}$ is still uniformly distributed on the set of all plane binary trees with size $n+1$. It turns out that the measure $\nu_{\tau_n}$, which we call the leaf-growth measure, is noticeably different from the uniform measure on the leaves of the tree $\tau_n$. 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.

• The paper introduces a novel leaf-growth measure that deviates from uniformity, concentrating growth on approximately n^0.8038 leaves in random binary trees.
• It employs asymptotic analysis, Poissonization, and renewal theory to show that the scaling limit converges to a probability measure on the Brownian CRT with fractal dimension 6(2‑√3).
• The findings offer deep insights into multifractal growth processes, setting the stage for further exploration in theoretical and applied network models.

An Analysis of "Where do (random) trees grow leaves?"

The paper "Where do (random) trees grow leaves?" investigates a probabilistic model for the growth of random binary trees. The model, inspired by the work of Luczak and Winkler, involves appending a "cherry" to a uniformly chosen leaf, resulting in a new tree that remains uniformly distributed among trees of the new size. This approach introduces a particular measure called the "leaf-growth measure," denoted as $\nu_{\tau_n}$, which deviates significantly from the uniform measure on a tree's leaves as the tree size $n$ increases.

Key Contributions and Findings

1. Leaf-Growth Measure $\nu_{\tau_n}$: The authors construct and analyze the measure $\nu_{\tau_n}$, which assigns different probabilities to the leaves of a tree. Unlike a uniform distribution, this measure becomes heavily supported on a subset of the leaves, specifically a set of size approximately $n^{0.8038}$ as $n \to \infty$.
2. Scaling Limit in the Continuum: In the continuum limit, the growth measure converges to a probability measure on the Brownian Continuum Random Tree (CRT), providing a fractal dimension of $6(2 - \sqrt{3})$. This result is supported by a comprehensive calculation of the multifractal spectrum of the measure.
3. Theoretical Implications: The results contribute to the understanding of random growth processes on trees by constructing the scaling limit of the discrete leaf-growth measure. This construction enables future theoretical exploration of diffusion limits of these procedures and their implications for related random structures.
4. Numerical and Analytical Techniques: The paper employs a combination of asymptotic analysis, Poissonization, and renewal theory to rigorously establish the properties of these growth measures. For example, it meticulously shows how the typical leaf-growth measure mass decays like $n^{-0.8038}$.

Practical and Theoretical Implications

• Fractal Nature and Multifractality: The revealed multifractal nature of the measures shows that different moments behave differently—a higher moment is not governed by the typical behavior but by a broader spectrum of growth behavior.
• Applications to Random Trees and Networks: The insights provided can be applied to a range of natural and engineered systems, including biological systems and computer networks, where branching structures are prevalent.
• Connections to Other Growth Models: While the current focus is on binary trees, extensions to more general $d$-ary trees could deepen the understanding of these stochastic couplings and tie into existing results such as those for Rémy's algorithm, which maintains a tree's uniform distribution under different assumptions.

Speculation on Future Developments

The framework established is a stepping stone towards understanding the dynamics of random structures that evolve via leaf growth. This could, in speculative future work, be expanded to more general classes of trees and networks. The notion of non-trivial stationary dynamics could also emerge in the limit, analogous to Markov processes, enabling further mathematical and algorithmic exploration in parallel with stochastic domination and other probabilistic tools.

Moreover, the authors conjecture that a consistent Markovian growth process could be defined in this framework, offering intriguing possibilities for modeling long-range interactions in evolving networks and providing models that capture complex dynamics in theoretical biology, economics, and beyond.

In conclusion, the paper by Caraceni et al. provides an in-depth exploration of the growth processes in random binary trees, revealing rich fractal behavior and setting the stage for future investigative pathways in understanding diffusive and deterministic dynamics in stochastic tree structures.