Limiting distributions for the number of inversions in labelled tree families

We consider so-called simple families of labelled trees, which contain, e.g., ordered, unordered, binary and cyclic labelled trees as special instances, and study the global and local behaviour of the number of inversions. In particular we obtain limiting distribution results for the total number of inversions as well as the number of inversions induced by the node labelled j in a random tree of size n.