  We consider the Ising model on a general tree under various boundary
conditions: all plus, free and spin-glass. In each case, we determine when the
root is influenced by the boundary values in the limit as the boundary recedes
to infinity. We obtain exact capacity criteria that govern behavior at critical
temperatures. For plus boundary conditions, an $L^3$ capacity arises. In
particular, on a spherically symmetric tree that has $n^{\alpha}b^n$ vertices
at level $n$ (up to bounded factors), we prove that there is a unique Gibbs
measure for the ferromagnetic Ising model at the relevant critical temperature
if and only if $\alpha\le1/2$. Our proofs are based on a new link between
nonlinear recursions on trees and $L^p$ capacities.
