  We consider the zero-temperature single-spin-flip dynamics of the
random-field Ising model on a Bethe lattice in the presence of an external
field h. We derive the exact self-consistent equations to determine the
distribution Prob(s) of avalanche sizes s, as the external field increases from
large negative to positive values. We solve these equations explicitly for a
rectangular distribution of the random fields for a linear chain and the Bethe
lattice of coordination number z=3, and show that in these cases, Prob(s)
decreases exponentially with s for large s for all h on the hysteresis loop. We
found that for z >3 and for small disorder, the magnetization shows a first
order discontinuity for several continuous and unimodel distributions of random
fields. The avalanche distribution Prob(s) varies as s^{-3/2} for large s near
the discontinuity.
