On the (2,2)-domination number of trees

Let γ(G) and ${\displaystyle {\gamma }_{2,2}\left(G\right)}$ denote the domination number and (2,2)-domination number of a graph G, respectively. In this paper, for any nontrivial tree T, we show that ${\displaystyle \frac{2(\gamma \left(T\right)+1)}{3}\le {\gamma }_{2,2}\left(T\right)\le 2\gamma \left(T\right)}$. Moreover, we characterize all the trees achieving the equalities.