  We present $O(\log^2 \log n)$ time 3-coloring, maximal independent set and
maximal matching algorithms for trees in the Massively Parallel Computation
(MPC) model. Our algorithms are deterministic, apply to arbitrary-degree trees
and work in the low-space MPC model, where local memory is $O(n^\delta)$ for
$\delta \in (0,1)$ and global memory is $O(m)$. Our main result is the
3-coloring algorithm, which contrasts the randomized, state-of-the-art
4-coloring algorithm of Ghaffari, Grunau and Jin [DISC'20]. The maximal
independent set and maximal matching algorithms follow in $O(1)$ time after
obtaining the coloring. The key ingredient of our 3-coloring algorithm is an
$O(\log^2 \log n)$ time adaptation of the rake-and-compress tree decomposition
used by Chang and Pettie [FOCS'17], and established by Miller and Reif. When
restricting our attention to trees of constant degree, we bring the runtime
down to $O(\log \log n)$.
