  We analyze kinetically constrained 0-1 spin models (KCSM) on rooted and
unrooted trees of finite connectivity. We focus in particular on the class of
Friedrickson-Andersen models FA-jf and on an oriented version of them. These
tree models are particularly relevant in physics literature since some of them
undergo an ergodicity breaking transition with the mixed first-second order
character of the glass transition. Here we first identify the ergodicity regime
and prove that the critical density for FA-jf and OFA-jf models coincide with
that of a suitable bootstrap percolation model. Next we prove for the first
time positivity of the spectral gap in the whole ergodic regime via a novel
argument based on martingales ideas. Finally, we discuss how this new technique
can be generalized to analyze KCSM on the regular lattice $\mathbb{Z}^d$.
