where for t = N in the sum over t , we define k$_{t}$$_{+1}$ ≡ k$_{1}$ . For a given r , the numerator of 〈 r,r +1 〉 〈 r,N +1 〉〈 N +1 ,r +1 〉 is a spinor product for two adjacent points. Then we can remove 〈 r, r + 1 〉 in 〈 r,r +1 〉 〈 r,N +1 〉〈 N +1 ,r +1 〉 and 〈 1 i 〉 4 〈 12 〉〈 23 〉 ... 〈 N 1 〉 simultaneously. Then 〈 r, r + 1 〉 in the denominator of 〈 1 i 〉 4 〈 12 〉〈 23 〉 ... 〈 N 1 〉 is replaced by 〈 r, N + 1 〉〈 N + 1 , r + 1 〉 . After this replacement, 〈 1 i 〉 4 〈 12 〉〈 23 〉 ... 〈 N 1 〉 becomes 〈 1 i 〉 4 〈 12 〉〈 23 〉 ... 〈 r,N +1 〉〈 N +1 ,r +1 〉 ... 〈 N 1 〉 which is the ( N + 1)-gluon MHV tree amplitude with the permutation 1, 2,..., r , N + 1, r + 1,..., N . Thus 〈 r,r +1 〉 〈 r,N +1 〉〈 N +1 ,r +1 〉 just insert a gluon with momentum k$_{N}$$_{+1}$ between the two gluons r and r + 1. For a given l (1 < l < i ), the sum over r for 1 ≤ r ≤ l − 1 becomes sum over all the possible insertions between 1 and l . For i < l ≤ N , the sum over r for i ≤ r ≤ l − 1 becomes sum over all the possible insertions between i and l . After a similar discussion, 〈 t,t +1 〉 〈 t,N +1 〉〈 N +1 ,t +1 〉 insert a gluon with momentum k$_{N}$$_{+1}$ at the positions between l , i for 1 < l < i and between i , 1 for i < l ≤ N . The two 14 where we define t + 1 = 1 for t = N . The amplitude then becomes For i < l ≤ N , between 1, l and l , i respectively. For 1 < l < i , A (1 − g , 2 + g , ..., i − g , ..., N + g , ( N + 1) + $_{h}$) = ig N − 2 ( κ 2 ) ( ∑ 1 <l<i s$_{l,N}$$_{+1}$ l − 1 ∑ r =1 i − 1 ∑ t = l + ∑ i<l ≤ N s$_{l,N}$$_{+1}$ l − 1 ∑ r = i N ∑ t = l ) · 〈 1 i 〉 4 〈 12 〉〈 23 ... 〈 N 1 〉 〈 r, r + 1 〉 〈 r, N + 1 〉〈 N + 1 , r + 1 〉 〈 t, t + 1 〉 〈 t, N + 1 〉〈 N + 1 , t + 1 〉 , (20) 〈 1 l 〉 〈 1 , N + 1 〉〈 N + 1 , l 〉 = l − 1 ∑ r =1 〈 r, r + 1 〉 〈 r, N + 1 〉〈 N + 1 , r + 1 〉 , 〈 li 〉 〈 l, N + 1 〉〈 N + 1 , i 〉 = i − 1 ∑ t = l 〈 t, t + 1 〉 〈 t, N + 1 〉〈 N + 1 , t + 1 〉 . (18) 〈 il 〉 〈 i, N + 1 〉〈 N + 1 , l 〉 = l − 1 ∑ r = i 〈 r, r + 1 〉 〈 r, N + 1 〉〈 N + 1 , r + 1 〉 , 〈 l 1 〉 〈 l, N + 1 〉〈 N + 1 , 1 〉 = N ∑ t = l 〈 t, t + 1 〉 〈 t, N + 1 〉〈 N + 1 , t + 1 〉 , (19)