  In the Maker-Breaker domination game played on a graph $G$, Dominator's goal
is to select a dominating set and Staller's goal is to claim a closed
neighborhood of some vertex. We study the cases when Staller can win the game.
If Dominator (resp., Staller) starts the game, then $\gamma_{\rm SMB}(G)$
(resp., $\gamma_{\rm SMB}'(G)$) denotes the minimum number of moves Staller
needs to win. For every positive integer $k$, trees $T$ with $\gamma_{\rm
SMB}'(T)=k$ are characterized and a general upper bound on $\gamma_{\rm SMB}'$
is proved. Let $S = S(n_1,\dots, n_\ell)$ be the subdivided star obtained from
the star with $\ell$ edges by subdividing its edges $n_1-1, \ldots, n_\ell-1$
times, respectively. Then $\gamma_{\rm SMB}'(S)$ is determined in all the cases
except when $\ell\ge 4$ and each $n_i$ is even. The simplest formula is
obtained when there are at least two odd $n_i$s. If $n_1$ and $n_2$ are the two
smallest such numbers, then $\gamma_{\rm SMB}'(S(n_1,\dots, n_\ell))=\lceil
\log_2(n_1+n_2+1)\rceil$. For caterpillars, exact formulas for $\gamma_{\rm
SMB}$ and for $\gamma_{\rm SMB}'$ are established.
