Inversions in Split Trees and Conditional Galton–Watson Trees

Abstract

We studyI(T), the number of inversions in a treeTwith its vertices labelled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants ofI(T) have explicit formulas involving thek-total common ancestors ofT(an extension of the total path length). Then we considerXn, the normalized version ofI(Tn), for a sequence of treesTn. For fixedTn's, we prove a sufficient condition forXnto converge in distribution. As an application, we identify the limit ofXnfor completeb-ary trees. ForTnbeing split trees [16], we show thatXnconverges to the unique solution of a distributional equation. Finally, whenTn's are conditional Galton–Watson trees, we show thatXnconverges 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 significantly strengthen and broaden previous work by Panholzer and Seitz [46].