  The $3/5$-conjecture for the domination game states that the game domination
numbers of an isolate-free graph $G$ on $n$ vertices are bounded as follows:
$\gamma_g(G)\leq \frac{3n}5 $ and $\gamma_g'(G)\leq \frac{3n+2}5 $. Recent
progress have been done on the subject and the conjecture is now proved for
graphs with minimum degree at least $2$. One powerful tool, introduced by
Bujt\'as is the so-called greedy strategy for \D. In particular, using this
strategy, she has proved the conjecture for isolate-free forests without leafs
at distance $4$. In this paper, we improve this strategy to extend the result
to the larger class of weakly $S(K_{1,3})$-free forests, where a weakly
$S(K_{1,3})$-free forest $F$ is an isolate-free forest without induced
$S(K_{1,3})$, whose leafs are leafs of $F$ as well.
