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Using $(771.6 \pm 10.6) \times 10^6$ $B\bar{B}$ meson pairs recorded by theBelle experiment at the KEKB $e^+ e^-$ collider, we report the branchingfractions $\mathcal{B}(\bar{B}^0 \to D^{*+} \pi^-) = (2.62 \pm 0.02 \pm0.09)\times 10^{-3}$ and $\mathcal{B}(\bar{B}^0 \to D^{*+} K^-) = (2.22 \pm0.06 \pm 0.08)\times 10^{-4}$; the quoted uncertainties are statistical andsystematic, respectively. A measurement of the ratio of these branchingfractions is also presented, $\mathcal{R}_{K/\pi}=\mathcal{B}(\bar{B} \toD^{*+}K^-) /\mathcal{B}( \bar{B} \to D^{*+}\pi^-) = (8.41 \pm 0.24 \pm0.13)\times 10^{-2}$, where systematic uncertainties due to the $D^{*+}$ mesonreconstruction cancel out. Furthermore, we report a new QCD factorization testbased on the measured ratios for $\bar{B} \to D^{*+}h^-$ and $\bar{B} \toD^{*+}\ell^-\nu$ decays at squared momentum transfer values equivalent to themass of the $h = \pi$ or $K$ hadron. The parameters $|a_1(h)|$ are measured tobe $|a_1(\pi)|= 0.884 \pm 0.004 \pm 0.003 \pm 0.016$ and $|a_1(K)|=0.913 \pm0.019 \pm 0.008 \pm 0.013$, where the last uncertainties account for allexternal inputs. These values are approximately $15\%$ lower than thoseexpected from theoretical predictions. Subsequently, flavor $SU(3)$ symmetry istested by measuring the ratios for pions and kaons, $|a_1(K)|^2/|a_1(\pi)|^2=1.066 \pm 0.042 \pm 0.018 \pm 0.023$, as well as for differentparticle species. The ratio is consistent with unity and therefore no evidencefor $SU(3)$ symmetry breaking effects is found at the $5\%$ precision level.

Measurements of the branching fractions $\mathcal{B} (\bar{B}{}^0 \to  D^{*+} \pi^-)$ and $\mathcal{B} (\bar{B}{}^0 \to D^{*+} K^-)$ and tests of  QCD factorization