Inversions in split trees and conditional Galton--Watson trees

We study $I(T)$, the number of inversions in a tree $T$ with its vertices labeled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants of $I(T)$ have explicit formulas involving the $k$-total common ancestors of $T$ (an extension of the total path length). Then we consider $X_n$, the normalized version of $I(T_n)$, for a sequence of trees $T_n$. For fixed $T_{n}$'s, we prove a sufficient condition for $X_n$ to converge in distribution. As an application, we identify the limit of $X_n$ for complete $b$-ary trees. For $T_n$ being split trees, we show that $X_n$ converges to the unique solution of a distributional equation. Finally, when $T_n$'s are conditional Galton--Watson trees, we show that $X_n$ converges to a random variable defined in terms of Brownian excursions. By exploiting the connection between inversions and the total path length, we are able to give results that are stronger and much broader compared to previous work by Panholzer and Seitz.