  Bona [2007+] studied the distribution of ascents, plateaux and descents in
the class of Stirling permutations, introduced by Gessel and Stanley [1978].
Recently, Janson [2008+] showed the connection between Stirling permutations
and plane recursive trees and proved a joint normal law for the parameters
considered by Bona. Here we will consider generalized Stirling permutations
extending the earlier results of Bona and Janson, and relate them with certain
families of generalized plane recursive trees, and also $(k+1)$-ary increasing
trees. We also give two different bijections between certain families of
increasing trees, which both give as a special case a bijection between ternary
increasing trees and plane recursive trees. In order to describe the
(asymptotic) behaviour of the parameters of interests, we study three
(generalized) Polya urn models using various methods.
