To study the relationship between the global 2-rainbow domination number γgr2(T) and 2-rainbow domination number γr2(T) of a tree T, by analyzing the structure of graphs and using the methods of categorical discussion and reduction to absurdity, the trees T of diameter five with γgr2(T)=γr2(T)+1 is completely characterized, which solves the unsolved problem by Amjadi et al. (2017).