Let X be a homogeneous tree of degree q+1 (for q between 2 and infinity) and
let f be a complex function on X times X for which f(x,y) only depend on the
distance between x and y in X. Our main result gives a necessary and sufficient
condition for such a function to be a Schur multiplier on X times X. Moreover,
we find a closed expression for the Schur norm of f. As applications, we obtain
a closed expression for the completely bounded Fourier multiplier norm of the
radial functions on the free (non-abelian) group on N generators (for N between
2 and infinity) and of the spherical functions on the p-adic group PGL_2(Q_q)
for every prime number q.