Given a simple graph G with m edges, we are looking for a bijection f from E(G) to the integer set {k + 1, k + 2, . . . , k + m} such that the vertex sum of each vertex v, ϕ(v), defined as the sum of f (e) over all edges e incident to v is unique. If such a bijection f exists, we say G is k-shifted antimagic. This is a generalization of the antimagic graphs proposed by Hartsfield and Ringel [7]. In this paper, we proved that every tree of diameter four or five, except for two previous known examples, is k-shifted antimagic for every integer k.