  In this letter we derive new expressions for tree-level graviton amplitudes
in $\mathcal{N}=8$ supergravity from BCFW recursion relations combined with new
types of bonus relations. These bonus relations go beyond the famous $1/z^2$
behavior under a large BCFW shift, and use knowledge about certain zeroes of
graviton amplitudes in collinear kinematics. This extra knowledge can be used
in the context of global residue theorems by writing the amplitude in a special
form using canonical building blocks. In the NMHV case these building blocks
are dressed one-loop leading singularities, the same objects that appear in the
expansion of Yang-Mills amplitudes, where each term corresponds to an
$R$-invariant. Unlike other approaches, our formula is not an expansion in
terms of cyclic objects and does not manifest color-kinematics duality, but
rather preserves the permutational symmetry of its building blocks. We also
comment on the possible connection to Grassmannian geometry and give some
non-trivial evidence of such structure for graviton amplitudes.
