随机树递归：哪些不动点对应于树的有形集合？
Random tree recursions: which fixed points correspond to tangible sets of trees?
摘要：
设$\mathcal{B}$是一组有根树，其中包含一个从根开始的无限二叉树。这个集合满足一个树属于它的元性质当且仅当它的根有子树$u$和$v$，使得根在$\mathcal{B}$和$p$的子树属于它。该亚属性使得$p$满足一个不动点方程，该方程可以有多个解。其中一个解决方案是$p$，但其他解决方案的含义是什么？特别地，它们是Galton-Watson树落入满足相同元性质的其他集合的概率吗？我们创建了一个框架，提出这类问题，我们分类的解决方案，以固定点方程，根据他们是否承认概率解释。我们的证明使用脊分解的高尔顿-沃森树和布尔函数的分析。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
