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Note: Please see pdf for full abstract with equations. Let D = (V,A) be a digraph. A double Roman dominating function on a digraph D is a function ƒ :V → {0, 1, 2, 3} such that every vertex u for which ƒ(u) = 0 has an in-neighbor v for which ƒ(v) = 3 or at least two in-neighbors assigned 2 under ƒ, while if ƒ(u) = 1, then the vertex u must have at least one in-neighbor assigned 2 or 3. The weight of a double Roman dominating function is the value ƒ(V) = ∑v∈V ƒ(v). The minimum weight of a double Roman dominating function on a digraph D is called the double Roman domination number of D, denoted by γdR (D). This paper gives a descriptive characterization of oriented trees T satisfying γdR(T ) = 2 (n − Δ+) + 1. 2020 Mathematics Subject Classification: 05C20,05C69.