  We show the first conditionally optimal deterministic algorithm for
$3$-coloring forests in the low-space massively parallel computation (MPC)
model. Our algorithm runs in $O(\log \log n)$ rounds and uses optimal global
space. The best previous algorithm requires $4$ colors [Ghaffari, Grunau, Jin,
DISC'20] and is randomized, while our algorithm are inherently deterministic.
  Our main technical contribution is an $O(\log \log n)$-round algorithm to
compute a partition of the forest into $O(\log n)$ ordered layers such that
every node has at most two neighbors in the same or higher layers. Similar
decompositions are often used in the area and we believe that this result is of
independent interest. Our results also immediately yield conditionally optimal
deterministic algorithms for maximal independent set and maximal matching for
forests, matching the state of the art [Giliberti, Fischer, Grunau, SPAA'23].
In contrast to their solution, our algorithms are not based on derandomization,
and are arguably simpler.
