We study several statistical mechanical models on a general tree.
Particular attention is devoted to the classical Heisenberg models, where the
state space is the $d$-dimensional unit sphere and the interactions are
proportional to the cosines of the angles between neighboring spins. The
phenomenon of interest here is the classification of phase transition
(non-uniqueness of the Gibbs state) according to whether it is robust.
In many cases, including all of the Heisenberg and Potts models, occurrence of
robust phase transition is determined by the geometry (branching number) of the
tree in a way that parallels the situation with independent percolation and
usual phase transition for the Ising model. The critical values for robust
phase transition for the Heisenberg and Potts models are also calculated
exactly. In some cases, such as the $q \geq 3$ Potts model, robust phase
transition and usual phase transition do not coincide, while in other cases,
such as the Heisenberg models, we conjecture that robust phase transition and
usual phase transition are equivalent. In addition, we show that symmetry
breaking is equivalent to the existence of a phase transition, a fact believed
but not known for the rotor model on $\mathbb{Z}^2$ .