  We discuss possible parallel textures of the Yukawa coupling matrices in the
generaltwo-Higgs-doublet model (2HDM). In those textures the flavor changing
neutral currentsare naturally suppressed. Motivated by a phenomenologically
successful texturewith four texture zeros in the standard model, we propose a
predictive ansatz for the Yukawa coupling matrices with the same texture in the
general 2HDM. Compared with the six texture-zero based ansatz proposed by Cheng
and Sher, it is in a better agreement with the data of quark mixings and CP
violation. The four texture-zero based ansatz predicts a different hierarchy in
the Yukawa coupling matrix elements. As a consequence, in the lepton sector,
the related Yukawa couplings are less constrained by the experimental upper
bound of $\mu\to e\gamma$, which allows significantly larger predictions for
other processes. The contributions from neutral scalar interactions to the
lepton number violation decay modes $\ell\to \ell_{1}\ell_{2}\ell_{3}$ are
calculated in both ansatz. It is shown that the predictions from the four
texture-zero based ansatz could be two order of magnitude greater than that
from the six texture-zero based one. The branching ratio of $\mu\to 3e$ and
$\tau\to 3\mu$ can reach $7.5 \times 10^{-17}$ and $1.3\tiems 10^{-10}$
respectively. The predicted ratio of $Br(\mu\to 3e)/Br(\tau\to 3e)$ is also
larger and almost parameter independent. Those differences make the two ansatz
to be easily distinguished by the future experiments.
