Every tree is 3-equitable

A labeling of a graph is a function f from the vertex set to some subset of the natural numbers. The image of a vertex is called its label. We assign the label |f(u)?f(v)| to the edge incident with vertices u and v. In a k-equitable labeling the image of f is the set {0,1,2,…,k?1}. We require both the vertex labels and the edge labels to be as equally distributed as possible, i.e., if v_i denotes the number of vertices labeled i and e_i denotes the number of edges labeled i, we require |v_i?v_j|1 and |e_i?e_j|1 for every i,j in {0,1,2,…,k?1}. Equitable graph labelings were introduced by I. Cahit as a generalization for graceful labeling. We prove that every tree is 3-equitable.