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On the limiting distribution of the number of improper edges for random trees
Abstract
Improper edges were introduced by Shor to refine Cayley's formula for rooted labeled trees. Zeng established a connection between Shor's refinement and the Ramanujan polynomials. Let 𝒯n denote the set of rooted labeled trees on [n]={1,…,n}. We prove that the number of improper edges in a uniformly random tree in 𝒯n is asymptotically normal as n→∞, with mean and variance asymptotic to μn and σ2 n, respectively, where μ=e−2 and σ2=e2−3e+1. This phenomenon was observed by Chen, and the proof presented here was developed through human--AI collaboration.
Related
BibTeX
@misc{ma2026limiting,
title = {{On the limiting distribution of the number of improper edges for random trees}},
author = {Wennie W. J. Ma and Kathy Q. Ji},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26045},
eprint = {2609.26045},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26045},
}
RIS
TY - UNPB
TI - On the limiting distribution of the number of improper edges for random trees
AU - Wennie W. J. Ma
AU - Kathy Q. Ji
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26045
UR - https://arxiv.org/abs/2609.26045
AB - Improper edges were introduced by Shor to refine Cayley's formula for rooted labeled trees. Zeng established a connection between Shor's refinement and the Ramanujan polynomials. Let 𝒯_n denote the set of rooted labeled trees on [n]={1,…,n}. We prove that the number of improper edges in a uniformly random tree in 𝒯_n is asymptotically normal as n→∞, with mean and variance asymptotic to μn and σ^2 n, respectively, where μ=e−2 and σ^2=e^2−3e+1. This phenomenon was observed by Chen, and the proof presented here was developed through human--AI collaboration.
ER -