  In this paper we complete the proof of the "equivalence" of non-discrete
R-buildings of types A~_2 and C~_2, with, respectively, projective planes and
generalized quadrangles with non-discrete valuation, begun in previous paper of
the authors. We also complete the proof of the "equivalence" of an affine
building of rank 3 with a generalized polygon with discrete valuation (by
proving this for generalized hexagons). We also complement a result of the
second author by proving uniqueness up to scalar multiples of the weight
sequences of polygons with non-discrete valuation. As an application, we
produce some new explicitly defined non-discrete R-buildings, in particular a
class of type A~_2 with arbitrary residues.
