Stirling permutations were introduced by Gessel and Stanley. A Stirling permutation is a permutation of the multiset {1,1,2,2,dots,n,n} such that for each i, the elements occuring between the two occurrences of i are larger than i. Stirling permutations are naturally related to second order Eulerian numbers and to Stirling polynomials.
We will present generalized Stirling permutations and relate them with certain families of increasing trees. The close relationship between Stirling permutations and trees allows to analyze parameters in Stirling permutations by analyzing their counterparts in the corresponding tree family. Furthermore, several parameters in random Stirling permutations can be described by so-called Polya-Eggenberger urn models. We present several results obtained using combinatorial and probabilistic methods.