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A maximum sequence $S$ of vertices in a graph $G$, so that every vertex in$S$ has a neighbor which is independent, or is itself independent, from allprevious vertices in $S$, is called a Grundy dominating sequence. The Grundydomination number, $\gamma_{gr}(G)$, is the length of $S$. We show that for anyforest $F$, $\gamma_{gr}(F)=|V(T)|-|\mathcal{P}|$ where $\mathcal{P}$ is aminimum partition of the non-isolate vertices of $F$ into caterpillars in whichif two caterpillars of $\mathcal{P}$ have an edge between them in $F$, thensuch an edge must be incident to a non-leaf vertex in at least one of thecaterpillars. We use this result to show the strong product conjecture of B.Bre\v{s}ar, Cs. Bujt\'{a}s, T. Gologranc, S. Klav\v{z}ar, G. Ko\v{s}mrlj, B.Patk\'{o}s, Zs. Tuza, and M. Vizer, Dominating sequences in grid-like andtoroidal graphs, Electron. J. Combin. 23(4): P4.34 (2016), for all forests.Namely, we show that for any forest $G$ and graph $H$, $\gamma_{gr}(G \boxtimesH) = \gamma_{gr}(G) \gamma_{gr}(H)$. We also show that every connected graph$G$ has a spanning tree $T$ so that $\gamma_{gr}(G)\le \gamma_{gr}(T)$ and thatevery non-complete connected graph contains a Grundy dominating set $S$ so thatthe induced subgraph of $S$ contains no isolated vertices.

Grundy domination of forests and the strong product conjecture