Summary: Let \(G\) be a graph with vertex set \(V(G)\). A double Roman dominating function (DRDF) on a graph \(G\) is a function \(f:V(G)\longrightarrow\{0,1,2,3\}\) that satisfies the following conditions: (i) If \(f(v)=0\), then \(v\) must have a neighbor \(w\) with \(f(w)=3\) or two neighbors \(x\) and \(y\) with \(f(x)=f(y)=2\); (ii) If \(f(v)=1\), then \(v\) must have a neighbor \(w\) with \(f(w)\ge 2\). The weight of a DRDF \(f\) is the sum \(\sum_{v\in V(G)}f(v)\). The double Roman domination number equals the minimum weight of a double Roman dominating function on \(G\). A double Italian dominating function (DIDF) is a function \(f:V(G)\longrightarrow \{0,1,2,3\}\) having the property that \(f(N[u])\geq 3\) for every vertex \(u\in V(G)\) with \(f(u)\in \{0,1\}\), where \(N[u]\) is the closed neighborhood of \(v\). The weight of a DIDF \(f\) is the sum \(\sum_{v\in V(G)}f(v)\), and the minimum weight of a DIDF in a graph \(G\) is the double Italian domination number. In this paper we first present Nordhaus-Gaddum type bounds on the double Roman domination number which improved corresponding results given in \textit{N. J. Rad} and \textit{H. Rahbani} [ibid. 39, No. 1, 41--53 (2019; Zbl 1401.05224)]. Furthermore, we establish lower bounds on the double Roman and double Italian domination numbers of trees.