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We propose an approach to analyze the asymptotic behavior of P\'olya urnsbased on the contraction method. For this, a new combinatorial discrete timeembedding of the evolution of the urn into random rooted trees is developed. Adecomposition of these trees leads to a system of recursive distributionalequations which capture the distributions of the numbers of balls of eachcolor. Ideas from the contraction method are used to study such systems ofrecursive distributional equations asymptotically. We apply our approach to acouple of concrete P\'olya urns that lead to limit laws with normal limitdistributions, with non-normal limit distributions and with asymptotic periodicdistributional behavior.

Polya urns via the contraction method