Summary: For a graph \(G = (V, E)\), a double Roman dominating function (DRDF) \(f : V \rightarrow \{0, 1, 2, 3\}\) has the property that for every vertex \(v\in V\) with \(f(v)=0\), either there exists a neighbor \(u \in N(v)\), with \(f(u)=3\), or at least two neighbors \(x, y \in N(v)\) having \(f(x)=f(y)=2\), and every vertex with value 1 under \(f\) has at least a neighbor with value 2 or 3. The weight of a DRDF is the sum \(f(V)= \sum_{v \in V}f(v)\). A DRDF \(f\) is an independent double Roman dominating function (IDRDF) if the vertices with weight at least two form an independent set. The independent double Roman domination number \(i_{dR}(G)\) is the minimum weight of an IDRDF on \(G\). In this paper, we show that for every tree \(T \) with diameter at least three, \(i(T)+i_R(T)-\frac{s(T)}{2} + 1 \leq i_{dR}(T) \leq i(T)+i_{R} (T)+s(T)-2\), where \(i(T),i_{R}(T)\) and \(s(T)\) are the independent domination number, the independent Roman domination number and the number of support vertex of \(T\), respectively.