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 $\mathscr{T}_n$ denote the set of rooted labeled trees on $[n]=\{1,\ldots,n\}$. We prove that the number of improper edges in a uniformly random tree in $\mathscr{T}_n$ is asymptotically normal as $n\to\infty$, with mean and variance asymptotic to $\mu n$ and $\sigma^2 n$, respectively, where $\mu=e-2$ and $\sigma^2=e^2-3e+1$. This phenomenon was observed by Chen, and the proof presented here was developed through human--AI collaboration.