  Let $D$ be a 2-dimensional regular local ring and let $Q(D)$ denote the
quadratic tree of 2-dimensional regular local overrings of $D$. We examine the
Noetherian rings that are intersections of rings in $Q(D)$. To do so, we
describe the desingularization of projective models over $D$ both algebraically
in terms of the saturation of complete ideals and order-theoretically in terms
of the quadratic tree $Q(D)$.
