A Roman dominating function (or just RDF) on a graph \(G =(V, E)\) is a function \(f: V \longrightarrow \{0, 1, 2\}\) satisfying the condition that every vertex u for which \(f(u) = 0\) is adjacent to at least one vertex v for which \(f(v) = 2\). The weight of an RDF f is the value \(f(V(G))=\sum _{u \in V(G)}f(u)\). An RDF f can be represented as \(f=(V_0,V_1,V_2)\), where \(V_i=\{v\in V:f(v)=i\}\) for \(i=0,1,2\). An RDF \(f=(V_0,V_1,V_2)\) is called a locating Roman dominating function (or just LRDF) if \(N(u)\cap V_2\ne N(v)\cap V_2\) for any pair u, v of distinct vertices of \(V_0\). The locating Roman domination number \(\gamma _R^L(G)\) is the minimum weight of an LRDF of G. A vertex v of a graph G is called a locating Roman domination critical vertex (or just \(\gamma _R^L\) critical vertex) if \(\gamma _R^L(G-v)<\gamma _R^L(G)\). In this paper, we characterize all trees with no \(\gamma _R^L\) critical vertices.

图\（G =（V，E）\）上的罗马支配函数（或仅是RDF）是满足以下条件的函数\（f：V \ longrightarrow \ {0，1，2 \} \）ü为此\（F（U）= 0 \）是相邻于至少一个顶点v为哪些\（F（v）= 2 \） 。RDF f的权重是值\（f（V（G））= \ sum _ {u \ in V（G）} f（u）\）。RDF f可以表示为\（f =（V_0，V_1，V_2）\），其中\（V_i = \ {v \ in V：f（v）= i \} \）对于\（i = 0， 1,2 \）。RDF \（f =（V_0，V_1，V_2）\）如果\（N（u）\ cap V_2 \ ne N（v）\ cap V_2 \）对于\（V_0 \）的任意对u，  v是\（N（u）\ cap V_2 \ ne N（v）\ cap V_2 \），则称为定位罗马支配函数（或简称LRDF）。定位的罗马统治数\（\ gamma _R ^ L（G）\）是G的LRDF的最小权重。如果\（\ gamma _R ^ L（Gv）<\ gamma _R ^ L（G），则图G的顶点v称为定位罗马统治关键顶点（或仅称为\（\ gamma _R ^ L \）关键顶点）。 \）。在本文中，我们描述了没有\（\ gamma _R ^ L \）关键顶点的所有树。