  An efficient density matrix renormalization group (DMRG) algorithm is
presented for the Bethe lattice with connectivity $Z = 3$ and antiferromagnetic
exchange between nearest neighbor spins $s= 1/2$ or 1 sites in successive
generations $g$. The algorithm is accurate for $s = 1$ sites. The ground states
are magnetic with spin $S(g) = 2^g s$, staggered magnetization that persists
for large $g > 20$ and short-range spin correlation functions that decrease
exponentially. A finite energy gap to $S > S(g)$ leads to a magnetization
plateau in the extended lattice. Closely similar DMRG results for $s$ = 1/2 and
1 are interpreted in terms of an analytical three-site model.
