  Consider any locally checkable labeling problem $\Pi$ in \emph{rooted regular
trees}: there is a finite set of labels $\Sigma$, and for each label $x \in
\Sigma$ we specify what are permitted label combinations of the children for an
internal node of label $x$ (the leaf nodes are unconstrained). This formalism
is expressive enough to capture many classic problems studied in distributed
computing, including vertex coloring, edge coloring, and maximal independent
set.
  We show that the distributed computational complexity of any such problem
$\Pi$ falls in one of the following classes: it is $O(1)$, $\Theta(\log^* n)$,
$\Theta(\log n)$, or $\Theta(n)$ rounds in trees with $n$ nodes (and all of
these classes are nonempty). We show that the complexity of any given problem
is the same in all four standard models of distributed graph algorithms:
deterministic LOCAL, randomized LOCAL, deterministic CONGEST, and randomized
CONGEST model. In particular, we show that randomness does not help in this
setting, and complexity classes such as $\Theta(\log \log n)$ or
$\Theta(\sqrt{n})$ do not exist (while they do exist in the broader setting of
general trees).
  We also show how to systematically determine the distributed computational
complexity of any such problem $\Pi$. We present an algorithm that, given the
description of $\Pi$, outputs the round complexity of $\Pi$ in these models.
While the algorithm may take exponential time in the size of the description of
$\Pi$, it is nevertheless practical: we provide a freely available
implementation of the classifier algorithm, and it is fast enough to classify
many typical problems of interest.
