  While Flory theories provide an extremely useful framework for understanding
the behavior of interacting, randomly branching polymers, the approach is
inherently limited. Here we use a combination of scaling arguments and computer
simulations to go beyond a Gaussian description. We analyse distributions
functions for a wide variety of quantities characterising the tree
connectivities and conformations for the four different statistical ensembles,
which we have studied numerically in [Rosa and Everaers, J. Phys. A (2016,
published) and J. Chem. Phys. (2016, to appear)]: (a) ideal randomly branching
polymers, (b) $2d$ and $3d$ melts of interacting randomly branching polymers,
(c) $3d$ self-avoiding trees with annealed connectivity and (d) $3d$
self-avoiding trees with quenched ideal connectivity. In particular, we
investigate the distributions (i) $p_N(n)$ of the weight, $n$, of branches cut
from trees of mass $N$ by severing randomly chosen bonds; (ii) $p_N(l)$ of the
contour distances, $l$, between monomers; (iii) $p_N(\vec r)$ of spatial
distances, $\vec r$, between monomers, and (iv) $p_N(\vec r|l)$ of the
end-to-end distance of paths of length $l$. Data for different tree sizes
superimpose, when expressed as functions of suitably rescaled observables $\vec
x = \vec r/\langle r^2(N) \rangle$ or $x =l/\langle l(N) \rangle$. In
particular, we observe a generalised Kramers relation for the branch weight
distributions (i) and find that all the other distributions (ii-iv) are of
Redner-des Cloizeaux type, $q(\vec x) = C \, |x|^\theta\ \exp \left( -(K |x|)^t
\right)$. We propose a coherent framework, including generalised Fisher-Pincus
relations, relating most of the RdC exponents to each other and to the contact
and Flory exponents for interacting trees.
