  We prove the continuity of the extremal decomposition measure of the free
state of low temperature Potts models, and more generally of ferromagnetic
finite-spin models, on a regular tree, including general clock models. The
decomposition is supported on uncountably many inhomogeneous extremal states,
that we call glassy states. The method of proof provides explicit concentration
bounds on branch overlaps, which play the role of an order parameter for
typical extremals. The result extends to the counterpart of the free state
(called central state) in a wide range of models which have no symmetry,
allowing also the presence of sufficiently small field terms. Our work shows in
particular that the decomposition of central states into uncountably many
glassy states in finite-spin models on trees at low temperature is a generic
phenomenon, and does not rely on symmetries of the Hamiltonian.
