Lemma 5.1 Let T be an optimal tree with weights W1≤w,≤W2≤.≤ W. Then there is an optimal tree t, so that T2s two vertices with weights w, and w2 are brother nodes. Proof: Let T be an optimal tree with weights W1≤W≤W2≤.≤W The two weights wa Wb are on lowest level and they are brothers. We denoted by v. and vh. let vo be the father of 2b Wa≥W1,Wb2W2 Let la be the length of the path from root to va, and lb be the length of the path from root to vh b We obtain a new tree T2 by exchanging from leaf v, to v and from leafy, to v w(T1)-w(T2)=(WaW1)(2-1)+(Wb-W2)(l2-l2)20▪ Lemma 5.1 Let T1 be an optimal tree with weights w1w2w3 wk . Then there is an optimal tree T2 so that T2 ’s two vertices with weights w1 and w2 are brother nodes. ▪ Proof: Let T1 be an optimal tree with weights w1w2w3wk ▪ The two weights wa ,wb are on lowest level and they are brothers. We denoted by va and vb . Let v0 be the father of va , vb . ▪ waw1 , wbw2 . ▪ Let l a be the length of the path from root to va , and lb be the length of the path from root to vb . ▪ l a =lb ▪ We obtain a new tree T2 by exchanging from leaf v1 to va and from leaf v2 to vb . ▪ w(T1 )-w(T2 )=(wa -w1 )(l a -l 1 )+ (wb -w2 )(l a -l 2 )0