We comprehensively study the charged-Higgs contributions to the leptonic $B^-_q to ell bar u$ ($q=u,c$) and semileptonic $bar B to X_q ell bar u$ ($X_u=pi, rho; X_c=D,D^*$) decays in the type-III two-Higgs-doublet model (2HDM). We employ the Cheng-Sher ansatz to suppress the tree-level flavor-changing neutral currents (FCNCs) in the quark sector. When the strict constraints from the $Delta B=2$ and $bto s gamma$ processes are considered, parameters $chi^u_{tq}$ from the quark couplings and $chi^ell_ell$ from the lepton couplings dictate the leptonic and semileptonic $B$ decays. It is found that when the measured $B^-_uto tau bar u$ and indirect bound of $B^-_c to tau bar u$ obtained by LEP1 data are taken into account, $R(D)$ and $R(pi)$ can have broadly allowed ranges; however, the values of $R(rho)$ and $R(D^*)$ are limited to approximately the standard model (SM) results. We also find that the same behaviors also occur in the $tau$-lepton polarizations and forward-backward asymmetries ($A^{X_q,tau}_{FB}$) of the semileptonic decays, with the exception of $A^{D^*,tau}_{FB}$, for which the deviation from the SM due to the charged-Higgs effect is still sizable. In addition, the $q^2$-dependent $A^{pi,tau}_{FB}$ and $A^{D,tau}_{FB}$ can be very sensitive to the charged-Higgs effects and have completely different shapes from the SM.
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