  We give a bijective proof of the Aztec diamond theorem, stating that there
are $2^{n(n+1)/2}$ domino tilings of the Aztec diamond of order $n$. The proof
in fact establishes a similar result for non-intersecting families of $n+1$
Schr\"oder paths, with horizontal, diagonal or vertical steps, linking the grid
points of two adjacent sides of an $n\times n$ square grid; these families are
well known to be in bijection with tilings of the Aztec diamond. Our bijection
is produced by an invertible "combing" algorithm, operating on families of
paths without non-intersection condition, but instead with the requirement that
any vertical steps come at the end of a path, and which are clearly
$2^{n(n+1)/2}$ in number; it transforms them into non-intersecting families.
