In this paper, using generalized {polya} urn models we find the expected value of the size of a branch in recursive $k$-ary trees. We also find the expectation of the number of nodes of a given outdegree in a branch of such trees.

The primary subject of this thesis is the notion of an urn model and their applications to complex systems. It is demonstrated that several models of social, physical, and biological science are special cases of a large class of urn models that are then exactly solved. These models prescribe 2 or more urns with N balls distributed among them. Two balls are then drawn randomly and then redistrib...

We analyze a preferential urn model with randomness using the replica method. The preferential urn model is a stochastic model based on the concept “the rich get richer.” The replica analysis clarifies that the preferential urn model with randomness shows a fat-tailed occupation distribution. We discuss the intuitive picture of the urn model, a relationship between quenched and annealing system...