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.