  An imitation of 2d field theory is formulated by means of a model on the
hierarchic tree (with branching number close to one) with the same potential
and the free correlators identical to those of 2d ones. Such a model possesses
some features of original models for certain scale invariant theories. For the
case of 2d conformal models it is possible to derive exact results. The
renormalization group equation for the free energy is a reaction-diffusion
equation, which is noise-free KPZ equation with an additional linear term. For
the case of Liouville model and strings these models on trees may be naturally
expressed via the Random Energy Model. This correspondence is used to identify
the phase structure of strings for analytical continuation of DDK expressions.
A phase transition is found for spherical strings a bit below three dimensions.
