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 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)). The tree amplitude (15) and (16), with M = 2 can be reduced to 3 In this section, we let the gluons corresponding to a graviton take the momentum of the graviton for convenience. This is a little different from in the sections above where each gluon from a graviton take half of the momentum of the graviton. However, a redefinition of the momentum k → 1 $_{2}$k only contribute a constant factor which does not affect our discussions. 15 B. D$_{2}$ relations for tree amplitudes with two positive helicity gravitons, N − 2 positive helicity gluons and two negative helicity gluons A (1 − g , 2 + g , ..., i − g , ..., N + g , ( N + 1) + $_{h}$) = ig N − 2 ( κ 2 ) ∑ l ∈{ g $^{+}$} s$_{l,N}$$_{+1}$ ∑ P A N +2 $_{MHV}$( P ) , (21) A (1 − g , 2 + g , ..., i − g , ..., N + g , ( N + 1) + $_{h}$, ( N + 2) + $_{h}$) = A + B + C , (22)