Note on the Weighted Internal Path Length of b-ary Trees

In a recent paper Broutin and Devroye (2005) have studied the height of a class of edge-weighted random trees. This is a class of trees growing in continuous time which includes many well known trees as examples. In this paper we derive a limit theorem for the internal path length for this class of trees. The application of this limit theorem to concrete examples depends upon the possibility to obtain an expansion of the mean of the path length. For the proof we extend a limit theorem in Neininger and Rüschendorf (2004) to recursive sequences of random variables with continuous time parameter.