αα-labeling number of trees

In this paper, we prove that the αα-labeling number of trees T  , Tα⩽⌈r/2⌉nTα⩽⌈r/2⌉n where n=|E(T)|n=|E(T)| and r is the radius of T  . This improves the known result Tα⩽eO(nlogn) tremendously and this upper bound is very close to the upper bound Tα⩽nTα⩽n conjectured by Snevily. Moreover, we prove that a tree with n edges and radius r   decomposes KtKt for some t⩽(r+1)n2+1t⩽(r+1)n2+1.