Maker-Breaker domination game on trees when Staller wins

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