Outer independent Roman domination number of trees

‎A Roman dominating function (RDF) on a graph G=(V,E) is a function  f : V → {0, 1, 2}  such that every vertex u for which f(u)=0 is‎ ‎adjacent to at least one vertex v for which f(v)=2‎. ‎An RDF f is called‎‎an outer independent Roman dominating function (OIRDF) if the set of‎‎vertices assigned a 0 under f is an independent set‎. ‎The weight of an‎‎OIRDF is the sum of its function values over all vertices‎, ‎and the outer‎‎independent Roman domination number ΥoiR (G) is the minimum weight‎‎of an OIRDF on $G$‎. ‎In this paper‎, ‎we show that if T is a tree of order n ≥ 3 with s(T) support vertices‎, ‎then $gamma _{oiR}(T)leq min‎ {‎frac{5n}{6},frac{3n+s(T)}{4}}.$ Moreover‎, ‎we characterize the tress‎‎attaining each bound‎.