On the limiting distribution of the number of improper edges for random trees

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 ${\mathcal{T}}_n$ denote the set of rooted labeled trees on $[n]=\{1,\dots ,n\}$. We prove that the number of improper edges in a uniformly random tree in ${\mathcal{T}}_n$ is asymptotically normal as $n\to \mathrm{\infty }$, 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.