  Let $\mathcal{B}$ be the set of rooted trees containing an infinite binary
subtree starting at the root. This set satisfies the metaproperty that a tree
belongs to it if and only if its root has children $u$ and $v$ such that the
subtrees rooted at $u$ and $v$ belong to it. Let $p$ be the probability that a
Galton-Watson tree falls in $\mathcal{B}$. The metaproperty makes $p$ satisfy a
fixed-point equation, which can have multiple solutions. One of these solutions
is $p$, but what is the meaning of the others? In particular, are they
probabilities of the Galton-Watson tree falling into other sets satisfying the
same metaproperty? We create a framework for posing questions of this sort, and
we classify solutions to fixed-point equations according to whether they admit
probabilistic interpretations. Our proofs use spine decompositions of
Galton-Watson trees and the analysis of Boolean functions.
