  A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if
every vertex is adjacent to a vertex in $S$. The total domination number
$\gamma_t(G)$ is the minimum cardinality of a total dominating set of $G$. The
total domination subdivision number $\mbox{sd}_{\gamma_t}(G)$ of a graph $G$ is
the minimum number of edges that must be subdivided (where each edge in $G$ can
be subdivided at most once) in order to increase the total domination number.
Haynes et al. (Discrete Math. 286 (2004) 195--202) have given a constructive
characterization of trees whose total domination subdivision number is~$3$. In
this paper, we give new characterizations of trees whose total domination
subdivision number is 3.
