  The conjectured duality relating all-loop leading singularities of n-particle
N^(k-2)MHV scattering amplitudes in N=4 SYM to a simple contour integral over
the Grassmannian G(k,n) makes all the symmetries of the theory manifest. Every
residue is individually Yangian invariant, but does not have a local space-time
interpretation--only a special sum over residues gives physical amplitudes. In
this paper we show that the sum over residues giving tree amplitudes can be
unified into a single algebraic variety, which we explicitly construct for all
NMHV and N^2MHV amplitudes. Remarkably, this allows the contour integral to
have a "particle interpretation" in the Grassmannian, where higher-point
amplitudes can be constructed from lower-point ones by adding one particle at a
time, with soft limits manifest. We move on to show that the connected
prescription for tree amplitudes in Witten's twistor string theory also admits
a Grassmannian particle interpretation, where the integral over the
Grassmannian localizes over the Veronese map from G(2,n) to G(k,n). These
apparently very different theories are related by a natural deformation with a
parameter t that smoothly interpolates between them. For NMHV amplitudes, we
use a simple residue theorem to prove t-independence of the result, thus
establishing a novel kind of duality between these theories.
