  The solutions to a large class of semi-linear parabolic PDEs are given in
terms of expectations of suitable functionals of a tree of branching particles.
A sufficient, and in some cases necessary, condition is given for the
integrability of the stochastic representation, using a companion scalar PDE.
  In cases where the representation fails to be integrable a sequence of pruned
trees is constructed, producing a approximate stochastic representations that
in some cases converge, globally in time, to the solution of the original PDE.
