  We study the constraints of the generic two-Higgs-doublet model (2HDM)
type-III and the impacts of the new Yukawa couplings. For comparisons, we
revisit the analysis in the 2HDM type-II. To understand the influence of all
involving free parameters and to realize their correlations, we employ
$\chi$-square fitting approach by including theoretical and experimental
constraints, such as S, T, and U oblique parameters, the production of standard
model Higgs and its decay to $\gamma\gamma$, $WW^*/ZZ^*$, $\tau^+\tau^-$, etc.
The errors of analysis are taken at $68\%$, $95.5\%$, and $99.7\%$ confidence
levels. Due to the new Yukawa couplings being associated with
$\cos(\beta-\alpha)$ and $\sin(\beta -\alpha)$, we find that the allowed
regions for $\sin\alpha$ and $\tan\beta$ in the type-III model can be broader
when the dictated parameter $\chi_F$ is positive; however, for negative
$\chi_F$, the limits are stricter than those in the type-II model. By using the
constrained parameters, we find that the deviation from the SM in the $h\to
Z\gamma$ can be of ${\cal O}(10\%)$. Additionally, we also study the top-quark
flavor-changing processes induced at the tree level in the type-III model and
find that when all current experimental data are considered, we get $Br(t\to
c(h, H) )< 10^{-3}$ for $m_h=125.36$ and $m_H=150$ GeV and $Br(t\to cA)$
slightly exceeds $10^{-3}$ for $m_A =130$ GeV.
