  We study the portraits of isometries of rooted trees - the labelling of the
tree, at each vertex, by the permutation of its descendants - in terms of
languages. We characterize regularly branched self-similar groups in terms of
$\omega$-regular languages. We deduce the algorithmic decidability of some
problems, such as the comparison of regularly branched contracting groups, and
their orbit structure on the boundary of the rooted tree.
