A set of vertices in a graph \( G \) is a global dominating set of \( G \) if it dominates both \( G \) and its complement \( \overline{G} \). The minimum cardinality of a global dominating set of \( G \) is the global domination number of \( G \). We explore the effects of graph modifications (edge removal, vertex removal, and edge addition) on the global domination number. In particular, for each graph modification, we study the global domination stable trees, that is, the trees whose global domination number remains the same upon the modification. We characterize these stable trees having small global domination numbers.