Let f be a function that assigns labels from the set { 0 , 1 , 2 , 3 , 4 } to the vertices of a simple graph G . The active neighborhood A N ( v ) of a vertex v ∈ V ( G ) with respect to f is the set of all neighbors of v that are assigned non-zero values under f . The function f is a perfect triple Roman dominating function (PTRD-function) on G if for every vertex v ∈ V ( G ) with f ( v ) < 3 , we have ∑ u ∈ N [ v ] f ( u ) = | A N ( v ) | + 3 . The weight of a PTRD-function is the sum of its function values over the whole set of vertices, and the PTRD-number is the minimum weight of a PTRD-function on G . In this paper, we show that determining the PTRD-number is NP-complete even when restricted to bipartite graphs. Moreover, the exact values of the PTRD-number for paths and cycles are established. Moreover, we provide an upper bound for the PTRD-number for trees of order at least five and we characterize the extremal trees attaining this upper bound.