  Current experimental data allow the zero value for one neutrino mass, either
m_1 = 0 or m_3 = 0. This observation implies that a realistic neutrino mass
texture can be established by starting from the limit (a) m_1 = m_2 = 0 and m_3
\neq 0 or (b) m_1 = m_2 \neq 0 and m_3 = 0. In both cases, we may introduce a
particular perturbation which ensures the resultant neutrino mixing matrix to
be the tri-bimaximal mixing pattern or its viable variations with all entries
being formed from small integers and their square roots. We find that it is
natural to incorporate this kind of neutrino mass matrix in the minimal Type-II
seesaw model with only one heavy right-handed Majorana neutrino N in addition
to the SU(2)_L Higgs triplet \Delta_L. We show that it is possible to account
for the cosmological baryon number asymmetry in the m_3 =0 case via thermal
leptogenesis, in which the one-loop vertex correction to N decays is mediated
by \Delta_L and the CP-violating asymmetry of N decays is attributed to the
electron flavor.
