  The random split tree introduced by Devroye (1999) is considered. We derive a
second order expansion for the mean of its internal path length and furthermore
obtain a limit law by the contraction method. As an assumption we need the
splitter having a Lebesgue density and mass in every neighborhood of 1. We use
properly stopped homogeneous Markov chains, for which limit results in total
variation distance as well as renewal theory are used. Furthermore, we extend
this method to obtain the corresponding results for the Wiener index.
