  The presence of the triplet $\Delta_{L}$ in left-right symmetric theories
leads to type-II see-saw mechanism for the neutrino masses. In these models,
assuming a normal mass hierarchy for the heavy Majorana neutrinos, we derive a
lower bound on the mass of the lightest of heavy Majorana neutrino from the
leptogenesis constraint. From this bound we establish a consistent picture for
the hierarchy of heavy Majorana neutrinos in a class of left right symmetric
models in which we identify the neutrino Dirac mass matrix with that of
Fritzsch type charged lepton mass matrix. It is shown that these values are
compatible with the current neutrino oscillation data.
