  We present a considerably improved analysis of model-independent bounds on
new physics effects in non-leptonic tree-level decays of B-mesons. Our main
finding is that contributions of about $\pm 0.1 $ to the Wilson coefficient of
the colour-singlet operator $Q_2$ of the effective weak Hamiltonian and
contributions in the range of $\pm 0.5$ (both for real and imaginary part) to
$Q_1$ can currently not be excluded at the $90\%$ C.L.. Effects of such a size
can modify the direct experimental extraction of the CKM angle $\gamma$ by up
to $10^{\circ}$ and they could lead to an enhancement of the decay rate
difference $\Delta \Gamma_d$ of up to a factor of 5 over its SM value - a size
that could explain the D0 dimuon asymmetry. Future more precise measurements of
the semi-leptonic asymmetries $a_{sl}^q$ and the lifetime ratio $\tau (B_s) /
\tau (B_d)$ will allow to shrink the bounds on tree-level new physics effects
considerably. Due to significant improvements in the precision of the
non-perturbative input we update all SM predictions for the mixing obervables
in the course of this analysis, obtaining: $\Delta M_s = (18.77 \pm 0.86 ) \,
\mbox{ps}^{-1}$, $\Delta M_d = (0.543 \pm 0.029) \, \mbox{ps}^{-1}$, $\Delta
\Gamma_s = (9.1 \pm 1.3 ) \cdot 10^{-2} \, \mbox{ps}^{-1}$, $\Delta \Gamma_d =
(2.6 \pm 0.4 ) \cdot 10^{-3} \, \mbox{ps}^{-1}$, $a_{sl}^s = (2.06 \pm 0.18)
\cdot 10^{-5}$ and $a_{sl}^d = (-4.73 \pm 0.42) \cdot 10^{-4}$.
