  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 all
previous vertices in $S$, is called a Grundy dominating sequence. The Grundy
domination number, $\gamma_{gr}(G)$, is the length of $S$. We show that for any
forest $F$, $\gamma_{gr}(F)=|V(T)|-|\mathcal{P}|$ where $\mathcal{P}$ is a
minimum partition of the non-isolate vertices of $F$ into caterpillars in which
if two caterpillars of $\mathcal{P}$ have an edge between them in $F$, then
such an edge must be incident to a non-leaf vertex in at least one of the
caterpillars. 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 and
toroidal graphs, Electron. J. Combin. 23(4): P4.34 (2016), for all forests.
Namely, we show that for any forest $G$ and graph $H$, $\gamma_{gr}(G \boxtimes
H) = \gamma_{gr}(G) \gamma_{gr}(H)$. We also show that every connected graph
$G$ has a spanning tree $T$ so that $\gamma_{gr}(G)\le \gamma_{gr}(T)$ and that
every non-complete connected graph contains a Grundy dominating set $S$ so that
the induced subgraph of $S$ contains no isolated vertices.
