PreprintarXiv licencearXiv
The height of discrete-time critical beta-splitting trees
Abstract
We determine the asymptotic height of the discrete-time critical beta-splitting tree. Let Ln* denotes the height of the tree with n leaves, defined as the maximum graph distance from the root to a leaf. Then, Ln*/((log n)2) ⟶ Cht:=minθ>1 θ/(2{ψ(θ)+γ}) ≈ 0.976 almost surely and in Lp for every fixed p>1 as n → ∞. Here ψ is the digamma function and γ is Euler's constant. This answers \cite[Open Problem~4]{AldousJansonII} of Aldous and Janson.
Related
- Extremal subtrees of critical beta-splitting trees
- GPU-Accelerated Bregman Douglas-Rachford Splitting for Discrete Optimal Transport
- Convergence and compactness of discrete aggregation trees
- Critical discrete Hardy inequalities in Lp
- Absence of free-time minimizers for N-body problems and a splitting theorem
BibTeX
@misc{ma2026height,
title = {{The height of discrete-time critical beta-splitting trees}},
author = {Heng Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14329},
eprint = {2609.14329},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14329},
}
RIS
TY - UNPB
TI - The height of discrete-time critical beta-splitting trees
AU - Heng Ma
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.14329
UR - https://arxiv.org/abs/2609.14329
AB - We determine the asymptotic height of the discrete-time critical beta-splitting tree. Let L_n^* denotes the height of the tree with n leaves, defined as the maximum graph distance from the root to a leaf. Then, L_n^*/((log n)^2) ⟶ C_ht:=min_θ>1 θ/(2{ψ(θ)+γ}) ≈ 0.976 almost surely and in L^p for every fixed p>1 as n → ∞. Here ψ is the digamma function and γ is Euler's constant. This answers \cite[Open Problem~4]{AldousJansonII} of Aldous and Janson.
ER -