  A model of irrigation network, where lower branches must be thicker in order
to support the weight of the higher ones, was recently introduced in BSUN2 [4].
This leads to a countable family of ODEs, describing the thickness of every
branch, solved by backward induction. The present paper determines what kind of
measures can be irrigated with a finite weighted cost. Indeed, the boundedness
of the cost depends on the dimension of the support of the irrigated measure,
and also on the asymptotic properties of the ODE which determines the thickness
of branches.
