Total restrained domination numbers of trees

For a given connected graph G=(V,E)G=(V,E), a set Dtr⊆V(G)Dtr⊆V(G) is a total restrained dominating set   if it is dominating and both 〈Dtr〉〈Dtr〉 and 〈V(G)-Dtr〉〈V(G)-Dtr〉 do not contain isolate vertices. The cardinality of the minimum total restrained dominating set in GG is the total restrained domination number   and is denoted by γtr(G)γtr(G). In this paper we characterize the trees with equal total and total restrained dominating numbers and give a lower bound on the total restrained dominating number of a tree TT in terms of its order and the number of leaves of TT.