A note on global alliances in trees

For a graph G = (V,E), a set S ⊆ V is a dominating set if every vertex in V - S has at least a neighbor in S. A dominating set S is a global offensive (respectively, defensive) alliance if for each vertex in V - S (respectively, in S) at least half the vertices from the closed neighborhood of v are in S. The domination number γ (G) is the minimum cardinality of a dominating set of G, and the global offensive alliance number γo(G) (respectively, global defensive alliance number γa(G)) is the minimum cardinality of a global offensive alliance (respectively, global deffensive alliance) of G. We show that if T is a tree of order n, then γo(T) ≤ 2γ (T) - 1 and if n ≥ 3, then γo(T) ≤ 3/2?a(T) ? 1. Moreover, all extremal trees attaining the first bound are characterized.