Total outer-connected domination numbers of trees

Let G=(V,E)G=(V,E) be a graph without an isolated vertex. A set D⊆V(G)D⊆V(G) is a total dominating set   if DD is dominating, and the induced subgraph G[D]G[D] does not contain an isolated vertex. The total domination number of GG is the minimum cardinality of a total dominating set of GG. A set D⊆V(G)D⊆V(G) is a total outer-connected dominating set   if DD is total dominating, and the induced subgraph G[V(G)−D]G[V(G)−D] is a connected graph. The total outer-connected domination number of GG is the minimum cardinality of a total outer-connected dominating set of GG. We characterize trees with equal total domination and total outer-connected domination numbers. We give a lower bound for the total outer-connected domination number of trees and we characterize the extremal trees.