This paper reviews P´olya urn models and their connection to random trees. Basic results are presented, together with proofs that underly the historical evolution of the accompanying thought process. Extensions and generalizations are given according to chronology: • P´olya-Eggenberger’s urn • Bernard Friedman’s urn • Generalized P´olya urns • Extended urn schemes • Invertible urn schemes ...

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.

Dynamical urn models, such as the Ehrenfest model, have played an important role in the early days of statistical mechanics. Dynamical many-urn models generalize the former models in two respects: the number of urns is macroscopic, and thermal effects are included. These many-urn models are exactly solvable in the mean-field geometry. They allow analytical investigations of the characteristic f...

In this work we discuss two urn models with general weight sequences (A,B) associated to them, A = (αn)n∈N andB = (βm)m∈N, generalizing two well known Pólya-Eggenberger urn models, namely the so-called sampling without replacement urn model and the OK Corral urn model. We derive simple explicit expressions for the distribution of the number of white balls, when all black have been drawn, and ob...

IV. Automata and generalized urn logic 11 A. Partition logic 13 B. Generalized urn models 13 C. Automaton models 13 D. Proof of logical equivalence 14 1. Direct construction of automaton models from generalized urn models 14 2. Direct construction of generalized urn models from automaton models 14 3. Schemes using dispersion-free states 14 4. Example 1: The generalized urn logic L12 16 5. Examp...

In this work we are concerned with so-called Pólya-Eggenberger urn models, which in the simplest case of two colors can be described as follows. At the beginning, the urn contains n white and m black balls. At every step, we choose a ball at random from the urn, examine its color and put it back into the urn and then add/remove balls according to its color by the following rules: if the ball is...