  We present $O(\log\log n)$ round scalable Massively Parallel Computation
algorithms for maximal independent set and maximal matching, in trees and more
generally graphs of bounded arboricity, as well as for constant coloring trees.
Following the standards, by a scalable MPC algorithm, we mean that these
algorithms can work on machines that have capacity/memory as small as
$n^{\delta}$ for any positive constant $\delta<1$. Our results improve over the
$O(\log^2\log n)$ round algorithms of Behnezhad et al. [PODC'19]. Moreover, our
matching algorithm is presumably optimal as its bound matches an
$\Omega(\log\log n)$ conditional lower bound of Ghaffari, Kuhn, and Uitto
[FOCS'19].
