  We provide a new set of on-shell recursion relations for tree-level
scattering amplitudes, which are valid for any non-trivial theory of massless
particles. In particular, we reconstruct the scattering amplitudes from (a
subset of) their poles and zeroes. The latter determine the boundary term
arising in the BCFW-representation when the amplitudes do not vanish as some
momenta are taken to infinity along some complex direction. Specifically, such
a boundary term can be expressed as a sum of products of two on-shell
amplitudes with fewer external states and a factor dependent on the location of
the relevant zeroes and poles. This allows us to recast the amplitudes to have
the standard BCFW-structure, weighted by a simple factor dependent on a subset
of zeroes and poles of the amplitudes. We further comment on the physical
interpretation of the zeroes as a particular kinematic limit in the
complexified momentum space. The main implication of the existence of such
recursion relations is that the tree-level approximation of any consistent
theory of massless particles can be fully determined just by the knowledge of
the corresponding three-particle amplitudes.
