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 eq 0 or (b) m_1
= m_2 eq 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.
