  We argue that the field theory that descibes randomly branched polymers is
not generally conformally invariant in two dimensions at its critical point. In
particular, we show (i) that the most natural formulation of conformal
invariance for randomly branched polymers leads to inconsistencies; (ii) that
the free field theory obtained by setting the potential equal to zero in the
branched polymer field theory is not even classically conformally invariant;
and (iii) that numerical enumerations of the exponent $\theta (\alpha )$,
defined by $T_N(\alpha )\sim \lambda^NN^{-\theta (\alpha ) +1}$, where
$T_N(\alpha )$ is number of distinct configuratations of a branched polymer
rooted near the apex of a cone with apex angel $\alpha$, indicate that $\theta
(\alpha )$ is not linear in $1/\alpha$ contrary to what conformal invariance
leads one to expect.
