Size-biased and conditioned random splitting trees

Random splitting trees share the striking independence properties of the continuous time binary Galton-Watson tree. They can be represented by Poisson point processes and their contour processes are strong Markov processes. Here we study splitting trees conditioned on extinction, respectively non-extinction as well as size-biased splitting trees. We give explicit probabilistic constructions of those trees by decomposing them into independent parts along a distinguished line of descent. The size-biased trees are shown to have stationary contour processes. Splitting trees are related to M/G/1-queuing systems which allows to translate the results on the trees into statements on the queues.