Tree polymers are simplifications of 1+1 dimensional
lattice polymers made up of polygonal paths of a (nonrecombining) binary
tree having random path probabilities. As in the case of lattice polymers, the
path probabilities are (normalized) products of i.i.d. positive weights. The a.s.
probability laws of these paths are of interest under weak and strong types of disorder.
Some recent results, speculation and conjectures will be presented for this class of models
under both weak and strong disorder conditions. In particular results are included that
suggest an explicit formula for the asymptotic variance of the ``free end'' under strong disorder.
This is based on joint work with Stanley Williams and Torrey Johnson.
