A. D$_{2}$ relations for tree amplitudes with one positive helicity graviton, N − 2 positive helicity gluons and two negative helicity gluons The tree amplitudes with two gluons of negative helicity and other legs of positive helicity can be expressed by[28, 29] where In M = 1 case where all the legs take positive helicity except 1 and i , the amplitude is reduced to In the equation above, 〈 1 i 〉 4 〈 12 〉〈 23 〉 ... 〈 N 1 〉 is just the MHV amplitude for N gluons with the per- mutation 1 , ..., N . Using (A5), we have 〈 l, N + 1 〉 [ l, N + 1] = s$_{l,N}$$_{+1}$ . With the eikonal identity(A9), 〈 1 l 〉 〈 1 ,N +1 〉〈 N +1 ,l 〉 and 〈 li 〉 〈 l,N +1 〉〈 N +1 ,i 〉 can split into sums of terms over all the legs 13 A (1 − g , 2 + g , ..., i − g , ..., N + g , ( N + 1) + $_{h}$) = ig N − 2 ( − κ 2 ) 〈 1 i 〉 4 〈 12 〉〈 23 ... 〈 N 1 〉 ∑ l ∈{ g $^{+}$} 〈 l 1 〉〈 li 〉 [ l, N + 1] 〈 N + 1 , 1 N + 1 , i 〉〈 l, N + 1 〉 = ig N − 2 ( κ 2 ) ∑ l ∈{ g $^{+}$} 〈 1 i 〉 4 〈 12 〉〈 23 ... 〈 N 1 〉 〈 l, N + 1 〉 [ l, N + 1] × 〈 1 l 〉 〈 1 , N + 1 〉〈 N + 1 , l 〉 〈 li 〉 〈 l, N + 1 〉〈 N + 1 , i 〉 . (17) S ( i, j, { h $^{+}$} , { g $^{+}$} ) =   ∏ m ∈{ h $^{+}$} d da$_{m}$   × ∏ l ∈{ g $^{+}$} exp   ∑ n$_{1}$ ∈{ h $^{+}$} a$_{n}$$_{1}$ 〈 li 〉〈 lj 〉 [ ln$_{1}$ ] 〈 n$_{1}$i n$_{1}$j 〉〈 ln$_{1}$ 〉 × exp   ∑ n$_{2}$ ∈{ h $^{+}$} ,n$_{2}$ = n$_{1}$ a$_{n}$$_{2}$ 〈 n$_{1}$i 〉〈 n$_{1}$j 〉 [ n$_{1}$n$_{2}$ ] n$_{2}$i n$_{2}$j 〉〈 n$_{1}$n$_{2}$ 〉 exp [ ... ]     ∣ a$_{j}$ =0 . (16) A (1 − g , 2 + g , ..., i − g , ..., N + g , ( N + 1) + $_{h}$, ..., ( N + M ) + $_{h}$) = ig N − 2 ( − κ 2 ) M 〈 1 i 〉 4 〈 12 〉〈 23 ... 〈 N 1 〉 S (1 , i, { h $^{+}$} , { g $^{+}$} ) , (15)