论文标题
随机树的自动形态数量的分布
The distribution of the number of automorphisms of random trees
论文作者
论文摘要
我们研究两种不同类型的随机树的自动形态组的大小：加尔顿 - 沃森树和生根的pólya树。在这两种情况下，我们都证明它渐近地遵循对数正态分布，并为自动形态群体大小的对数的平均值和方差提供渐近公式。虽然Galton的证明 - Watson树主要依赖于概率参数和对添加树功能的一般结果，但在生根的Pólya树的情况下使用生成功能。我们还展示了如何将结果扩展到一些未根的树。
We study the size of the automorphism group of two different types of random trees: Galton--Watson trees and rooted Pólya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic formulas for mean and variance of the logarithm of the size of the automorphism group. While the proof for Galton--Watson trees mainly relies on probabilistic arguments and a general result on additive tree functionals, generating functions are used in the case of rooted Pólya trees. We also show how to extend the results to some classes of unrooted trees.
