Let 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 . 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.

中文翻译：

---

随机树递归：哪些不动点对应于有形树集？

令其为包含从根开始的无限二叉子树的根树集合。当且仅当树的根有子u和v且以u和v为根的子树属于它时，此集合才满足该树所属的元属性。令p为Galton-Watson树落入的概率。元性质使p满足一个定点方程，该定点方程可以有多个解。这些解决方案之一是p，但是其他是什么意思呢？特别是，它们是否将高尔顿-沃森树的概率落入满足相同元属性的其他集合中？我们创建了一个提出此类问题的框架，并根据定点方程是否接受概率解释将其分类。我们的证明使用Galton-Watson树的脊柱分解和布尔函数分析。