  We consider the minimization of the NLS energy on a metric tree, either
rooted or unrooted, subject to a mass constraint. With respect to the same
problem on other types of metric graphs, several new features appear, such as
the existence of minimizers with positive energy, and the emergence of
unexpected threshold phenomena. We also study the problem with a radial
symmetry constraint that is in principle different from the free problem due to
the failure of the P\'olya-Szeg\H{o} inequality for radial rearrangements. A
key role is played by a new Poincar\'e inequality with remainder.
