preprint · arXiv (Cornell University) · 2024

Where do (random) trees grow leaves?

Alessandra Caraceni, Nicolas Curien, Robin Stephenson

Cited by 0Open access

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.

Cite this paper

APA 7

Caraceni, A., Curien, N., & Stephenson, R. (2024). Where do (random) trees grow leaves? arXiv (Cornell University). https://doi.org/10.48550/arxiv.2401.07891

MLA 9

Caraceni, Alessandra, et al. “Where do (random) trees grow leaves?” arXiv (Cornell University), 2024, https://doi.org/10.48550/arxiv.2401.07891.

Chicago

Caraceni, Alessandra, Nicolas Curien, and Robin Stephenson. 2024. “Where do (random) trees grow leaves?” arXiv (Cornell University). https://doi.org/10.48550/arxiv.2401.07891.

Harvard

Caraceni, A., Curien, N. and Stephenson, R. (2024) ‘Where do (random) trees grow leaves?’, arXiv (Cornell University). Available at: https://doi.org/10.48550/arxiv.2401.07891.

BibTeX

@misc{caraceni2024where,
  title = {{Where do (random) trees grow leaves?}},
  author = {Caraceni, Alessandra and Curien, Nicolas and Stephenson, Robin},
  journal = {arXiv (Cornell University)},
  year = {2024},
  doi = {10.48550/arxiv.2401.07891}
}

Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.