  A possible interplay of both terms in the type II see-saw formula is
illustrated by presenting a novel way to generate deviations from exact
bimaximal neutrino mixing. In type II see-saw mechanism with dominance of the
non-canonical SU(2)_L triplet term, the conventional see-saw term can give a
small contribution to the neutrino mass matrix. If the triplet term corresponds
to the bimaximal mixing scheme in the normal hierarchy, the small contribution
of the conventional see-saw term naturally generates non-maximal solar neutrino
mixing. Atmospheric neutrino mixing is also reduced from maximal, corresponding
to 1 - \sin^2 2 \theta_{23} of order 0.01. Also, small but non-vanishing U_{e3}
of order 0.001 is obtained. It is also possible that the \Delta m^2 responsible
for solar neutrino oscillations is induced by the small conventional see-saw
term. Larger deviations from zero U_{e3} and from maximal atmospheric neutrino
mixing are then expected. This scenario links the small ratio of the solar and
atmospheric \Delta m^2 with the deviation from maximal solar neutrino mixing.
We comment on leptogenesis in this scenario and compare the contributions to
the decay asymmetry of the heavy Majorana neutrinos as induced by themselves
and by the triplet.
