We present a new, explicit formula for all tree-level amplitudes in N=4 superYang-Mills. The formula is written as a certain contour integral of theconnected prescription of Witten's twistor string, expressed in link variables.A very simple deformation of the integrand gives directly the Grassmannianintegrand proposed by Arkani-Hamed et al. together with the explicit contour ofintegration. The integral is derived by iteratively adding particles to theGrassmannian integral, one particle at a time, and makes manifest both parityand soft limits. The formula is shown to be related to those given by Dolan andGoddard, and generalizes the results of earlier work for NMHV and N^2MHV to allN^(k-2)MHV tree amplitudes in N=4 super Yang-Mills.
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