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In this paper, we continue the study of the total domination game in graphsintroduced in [Graphs Combin. 31(5) (2015), 1453--1462], where the playersDominator and Staller alternately select vertices of $G$. Each vertex chosenmust strictly increase the number of vertices totally dominated, where a vertextotally dominates another vertex if they are neighbors. This process eventuallyproduces a total dominating set $S$ of $G$ in which every vertex is totallydominated by a vertex in $S$. Dominator wishes to minimize the number ofvertices chosen, while Staller wishes to maximize it. The game total dominationnumber, $\gamma_{\rm tg}(G)$, (respectively, Staller-start game totaldomination number, $\gamma_{\rm tg}'(G)$) of $G$ is the number of verticeschosen when Dominator (respectively, Staller) starts the game and both playersplay optimally. For general graphs $G$, sometimes $\gamma_{\rm tg}(G) >\gamma_{\rm tg}'(G)$. We show that if $G$ is a forest with no isolated vertex,then $\gamma_{\rm tg}(G) \le \gamma_{\rm tg}'(G)$. Using this result, wecharacterize the trees with equal total domination and game total dominationnumber.

Trees with Equal Total Domination and Game Total Domination Numbers