Some remarks on biased recursive trees

Abstract

The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform permutations, biased riffle shuffles. The statistics that are treated include the number of nodes with a given number of descendants, the depth of the tree, and the number of branches. The model yields the uniform recursive trees as a certain limit, some new results for the uniform case are obtained as well.

A study on random permutation graphs

“…Therefore, again permutation statistics that are related to descents can be understood via independent random variables, though with random words this time. This observation was previously used in [4] and was further developed in [21]. Also, see [20].…”