  We study boson stars in Brans Dicke gravity and use them to illustrate some
of the properties of three different mass definitions: the Schwarzschild mass,
the Keplerian mass and the Tensor mass. We analyse the weak field limit of the
solutions and show that only the Tensor mass leads to a physically reasonable
definition of the binding energy. We examine numerically strong field
$\omega=-1$ solutions and show how, in this extreme case, the three mass values
and the conserved particle number behave as a function of the central boson
field amplitude. The numerical studies imply that for $\omega=-1$, solutions
with extremal Tensor mass also have extremal particle number. This is a
property that a physically reasonable definition of the mass of a boson star
must have, and we prove analytically that this is true for all values of
$\omega$. The analysis supports the conjecture that the Tensor mass uniquely
describes the total energy of an asymptotically flat solution in BD gravity.
