The distribution of the maximum protection number in simply generated trees

Clemens Heuberger, Sarah J. Selkirk, Stephan Wagner
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Abstract

The protection number of a vertex $v$ in a tree is the length of the shortest path from $v$ to any leaf contained in the maximal subtree where $v$ is the root. In this paper, we determine the distribution of the maximum protection number of a vertex in simply generated trees, thereby refining a recent result of Devroye, Goh, and Zhao. Two different cases can be observed: if the given family of trees allows vertices of outdegree $1$ , then the maximum protection number is on average logarithmic in the tree size, with a discrete double-exponential limiting distribution. If no such vertices are allowed, the maximum protection number is doubly logarithmic in the tree size and concentrated on at most two values. These results are obtained by studying the singular behaviour of the generating functions of trees with bounded protection number. While a general distributional result by Prodinger and Wagner can be used in the first case, we prove a variant of that result in the second case.
简单生成树中最大保护数的分布
树中顶点 $v$ 的保护数是指从 $v$ 到包含在最大子树(其中 $v$ 为根)中任何叶子的最短路径的长度。本文确定了简单生成树中顶点最大保护数的分布,从而完善了 Devroye、Goh 和 Zhao 的最新成果。我们可以观察到两种不同的情况:如果给定的树族允许外度为 1$ 的顶点,那么最大保护数平均与树的大小成对数关系,具有离散的双指数极限分布。如果不允许有这样的顶点,则最大保护数是树大小的双对数,且最多集中在两个值上。这些结果是通过研究具有有界保护数的树的生成函数的奇异行为得到的。在第一种情况下,可以使用普罗丁格和瓦格纳的一般分布结果,而在第二种情况下,我们证明了该结果的一个变体。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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