标题： 伽顿-沃森过程，树的简单类型和金钦族 显示英文标题

标题： Galton-Watson processes, simple varieties of trees and Khinchin families

摘要： 在本文中，我们引入了一个统一的分析框架，将简单的树类、Bienayme-Galton-Watson 过程和 Khinchin 家族联系起来。 使用 Lagrange 的反演公式，我们推导出灭绝概率的新系数表达式，并将其重新解释为与 Lagrange 方程解的逆的定义域相关的边界现象。 这种观点揭示了组合模型和概率模型之间的结构桥梁，简化了经典论证并得到了新的结果。 它还导致了一种通过幂级数系数模拟 Galton-Watson 过程的计算高效方法。 显示英文摘要

摘要： In this note, we introduce a unified analytic framework that connects simple varieties of trees, Bienayme-Galton-Watson processes and Khinchin families. Using Lagrange's inversion formula, we derive new coefficient-based expressions for extinction probabilities and reinterpret them as boundary phenomena tied to the domain of the inverse of the solution to Lagrange's equation. This perspective reveals a structural bridge between combinatorial and probabilistic models, simplifying classical arguments and yielding new results. It also leads to a computationally efficient method for simulating Galton-Watson processes via power series coefficients.