  Panagiotou and Stufler recently proved an important fact on their way to
establish the scaling limits of random P\'olya trees: a uniform random P\'olya
tree of size $n$ consists of a conditioned critical Galton-Watson tree $C_n$
and many small forests, where with probability tending to one, as $n$ tends to
infinity, any forest $F_n(v)$, that is attached to a node $v$ in $C_n$, is
maximally of size $\vert F_n(v)\vert=O(\log n)$. Their proof used the framework
of a Boltzmann sampler and deviation inequalities.
  In this paper, first, we employ a unified framework in analytic combinatorics
to prove this fact with additional improvements for $\vert F_n(v)\vert$, namely
$\vert F_n(v)\vert=\Theta(\log n)$. Second, we give a combinatorial
interpretation of the rational weights of these forests and the defining
substitution process in terms of automorphisms associated to a given P\'olya
tree. Third, we derive the limit probability that for a random node $v$ the
attached forest $F_n(v)$ is of a given size. Moreover, structural properties of
those forests like the number of their components are studied. Finally, we
extend all results to other P\'olya structures.
