  In this paper, we generalize the Nguyen-Spradlin-Volovich-Wen (NSVW) tree
formula from the MHV sector to any helicity sector. We find a close connection
between the Permutohedron and the KLT relation, and construct a non-trivial
mapping between them, linking the amplitudes in the gauge and gravity theories.
The gravity amplitude can also be mapped from a determinant followed from the
matrix-tree theorem. Besides, we use the binary tree graphs to manifest its Lie
structure. In our tree formula, there is an evident Hopf algebra of the
permutation group behind the gravity amplitudes. Using the tree formula, we can
directly re-derive the soft/collinear limit of the amplitudes.
