  Three-body systems in two dimensions with zero-range interactions are
considered for general masses and interaction strengths. The problem is
formulated in momentum space and the numerical solution of the Schr\"odinger
equation is used to study universal properties of such systems with respect to
the bound-state energies. The number of universal bound states is represented
in a form of boundaries in a mass-mass diagram. The number of bound states is
strongly mass dependent and increases as one particle becomes much lighter than
the other ones. This behavior is understood through an accurate analytical
approximation to the adiabatic potential for one light particle and two heavy
ones.
