On trees attaining an upper bound on the total domination number

‎A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$‎. ‎The total domination number of a graph $G$‎, ‎denoted by $\gamma_t(G)$‎, ‎is~the minimum cardinality of a total dominating set of $G$‎. ‎Chellali and Haynes [{\it Total and paired-domination numbers of a tree,} AKCE International Journal of Graphs and Combinatorics 1 (2004)‎, ‎69--75] established the following upper bound on the total domination number of a tree in terms of the order and the number of support vertices‎, ‎$\gamma_t(T) \le (n+s)/2$‎. ‎We characterize all trees attaining this upper bound‎.‎