Restrained Italian domination in trees

Let be a graph. A subset D of V is a restrained dominating set if every vertex in is adjacent to a vertex in D and to a vertex in . The restrained domination number, denoted by , is the smallest cardinality of a restrained dominating set of G. A function is a restrained Italian dominating function on G if (i) for each vertex for which , it holds that , (ii) the subgraph induced by has no isolated vertices. The restrained Italian domination number, denoted by , is the minimum weight taken over all restrained Italian dominating functions of G. It is known that for any graph G. In this paper, we characterize the trees T for which , and we also characterize the trees T for which .