  Let $G=$ $\left( V,E\right) $ be a simple graph.\ A non-empty set $S
\subseteq V$ is called a global offensive alliance if $S$ is a dominating set
and for every vertex $v$ in $V-S$, at least half of the vertices from the
closed neighborhood of $v$ are in $S$. The global offensive alliance number is
the minimum cardinality of a global offensive alliance in $G$. In this paper,
we give a constructive characterization of trees having a unique minimum global
offensive alliance.
