  This article will review recent results on dimensional reduction for branched
polymers, and discuss implications for critical phenomena. Parisi and Sourlas
argued in 1981 that branched polymers fall into the universality class of the
Yang-Lee edge in two fewer dimensions. Brydges and I have proven in
[math-ph/0107005] that the generating function for self-avoiding branched
polymers in D+2 continuum dimensions is proportional to the pressure of the
hard-core continuum gas at negative activity in D dimensions (which is in the
Yang-Lee or $i \phi^3$ class). I will describe how this equivalence arises from
an underlying supersymmetry of the branched polymer model.
  -
  I will also use dimensional reduction to analyze the crossover of
two-dimensional branched polymers to their mean-field limit, and to show that
the scaling is given by an Airy function (the same as in [cond-mat/0107223]).
