  The universal three-body dynamics in ultra-cold binary gases confined to
one-dimensional motion are studied. The three-body binding energies and the (2
+ 1)-scattering lengths are calculated for two identical particles of mass $m$
and a different one of mass $m_1$, which interactions is described in the
low-energy limit by zero-range potentials. The critical values of the mass
ratio $m/m_1$, at which the three-body states arise and the (2 + 1)-scattering
length equals zero, are determined both for zero and infinite interaction
strength $\lambda_1$ of the identical particles. A number of exact results are
enlisted and asymptotic dependences both for $m/m_1 \to \infty$ and $\lambda_1
\to -\infty$ are derived. Combining the numerical and analytical results, a
schematic diagram showing the number of the three-body bound states and the
sign of the (2 + 1)-scattering length in the plane of the mass ratio and
interaction-strength ratio is deduced. The results provide a description of the
homogeneous and mixed phases of atoms and molecules in dilute binary quantum
gases.
