Some bounds on global alliances in trees

Given a simple graph G=(V,E), a subset S of the vertices is called a global defensive alliance if S is a dominating set and for every vertex v in S at least half of the vertices in the closed neighborhood of v are in S. Similarly, a subset S is called a global offensive alliance if S is a dominating set and for every vertex v not in S at least half of the vertices in the closed neighborhood of v are in S. In this paper, we study the minimum cardinality global defensive and global offensive alliances of complete k-ary trees. We also give a bound on the difference between these two parameters for general trees.