Limit laws for two distance-based indices in random recursive tree models

In this paper, we derive several results related to total path length and Sackin index in two classes of random recursive trees. A limiting distribution of the normalized version of the Sackin index is given by the contraction method in random recursive trees. Also, we show the normalized total path length converges in L² and almost surely to a limiting random variable in plane-oriented recursive trees via martingales.