  We propose an approach to analyze the asymptotic behavior of P\'olya urns
based on the contraction method. For this, a new combinatorial discrete time
embedding of the evolution of the urn into random rooted trees is developed. A
decomposition of these trees leads to a system of recursive distributional
equations which capture the distributions of the numbers of balls of each
color. Ideas from the contraction method are used to study such systems of
recursive distributional equations asymptotically. We apply our approach to a
couple of concrete P\'olya urns that lead to limit laws with normal limit
distributions, with non-normal limit distributions and with asymptotic periodic
distributional behavior.
