content_image/305426.jpg content_image/305427.jpg
gluons with momentum k$_{N}$$_{+1}$ just correspond to the left- and right-moving sectors of the graviton $^{3}$. Then 〈 1 i 〉 4 〈 12 〉〈 23 〉 ... 〈 N 1 〉 becomes 〈 1 i 〉 4 〈 12 〉〈 23 〉 ... 〈 r,N +1 〉〈 N +1 ,r +1 〉 ... 〈 t,N +1 〉〈 N +1 ,t 〉 ... 〈 N 1 〉 which is the MHV tree amplitude with N + 2 gluons. 

Thus, the amplitude can be considered as a sum of terms. In each term, there is an MHV tree amplitude for N + 2 gluons. Two of the N + 2 gluons take the momentum k$_{N}$$_{+1}$ . Then the amplitude satisfies the relation 

<img src='content_image/305426.jpg'>

Where for any given l in { g $^{+}$} , we sum over permutations P . P are the permutations in which the relative position of the N gluons is 1$_{g}$, 2$_{g}$,..., N$_{g}$ , one gluon corresponding to the graviton ( N + 1)$_{h}$ can be inserted at any position between 1$_{g}$ and l$_{g}$ , the other gluon corresponding to the graviton can be inserted at any position between l$_{g}$ and i$_{g}$ (See Fig. 1(a) and (b)). 

## B. D$_{2}$ relations for tree amplitudes with two positive helicity gravitons, N − 2 positive helicity gluons and two negative helicity gluons 

The tree amplitude (15) and (16), with M = 2 can be reduced to 

<img src='content_image/305427.jpg'>

