Trees with equal global offensive k-alliance and k-domination numbers

Let k ≥ 1 be an integer. A set S of vertices of a graph G = (V (G), E(G)) is called a global offensive k-alliance if |N(v) ∩ S| ≥ |N(v) - S| + k for every v ∈ V (G) - S, where N(v) is the neighborhood of v. The subset S is a k-dominating set of G if every vertex in V (G) - S has at least k neighbors in S. The global offensive k-alliance number [formula] is the minimum cardinality of a global offensive k-alliance in G and the k-domination number ϒ k(G) is the minimum cardinality of a k-dominating set of G. For every integer k ≥ 1 every graph G satisfies [formula]. In this paper we provide for k ≥ 2 a characterization of trees T with equal [formula] and ϒ k(T).