  We investigate the next-to-leading-colour (NLC) contributions to the colour
matrix in the fundamental and the colour-flow decompositions for tree-level
processes with all gluons, one quark pair and two quark pairs. By analytical
examination of the colour factors, we find the non-zero elements in the colour
matrix at NLC. At this colour order, together with the symmetry of the
phase-space, it is reduced from factorial to polynomial the scaling of the
contributing dual amplitudes as the number of partons participating in the
scattering process is increased. This opens a path to an accurate tree-level
matrix element generator of which all factorial complexity is removed, without
resulting to Monte Carlo sampling over colour.
