临界β-分裂随机树II：综述与开放问题
The Critical Beta-splitting Random Tree II: Overview and Open Problems
AI总结：
本文综述临界β-分裂随机树模型的结构理论与定量研究结果，围绕叶高CLT、随机叶极限边缘分布、离散模型嵌入连续时间模型CTCS(n)展开，强调其作为技术试验平台的价值及CTCS(∞)与β(2,1)合并关系等开放问题。
摘要：
在随机n叶根树的临界β-分裂模型中，分支从根开始递归地分裂为子分支，一个含m叶的分支分裂为含i叶和m-i叶的子分支的概率∝1/(i(m-i))。该模型（离散或连续版本）的结构理论与显式定量方面研究是活跃课题。许多结果有概率或分析等不同证明，因此该模型为技术的“比较与对比”讨论提供了试验平台。本文综述了标题相似的I、III、IV系列文章及相关文献中的结果，大多不重复已有证明，而是通过图形和启发式方法描绘“全景”并强调开放问题。我们的讨论围绕三类结果展开：（i）叶高存在中心极限定理（CLT），分析证明可扩展以对其他高度相关方面进行出人意料的精确分析；（ii）存在关于随机叶的极限边缘分布的显式描述，其图形表示本质上是生物系统发育分支图的格式；（iii）离散模型可嵌入连续时间模型CTCS(n)（n叶实值边长随机树），该模型更便于研究。家族(CTCS(n),n≥2)在“删除随机叶并修剪”操作下是一致的，这导致n增大时(CTCS(n),n≥2)的显式归纳构造，进而通过可交换分区形式化极限结构CTCS(∞)。仍存在许多开放问题，尤其是阐明CTCS(∞)与β(2,1)合并过程之间的关系。
英文摘要：
In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively (from the root) split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$. Study of structure theory and explicit quantitative aspects of this model (in discrete or continuous versions) is an active research topic. For many results there are different proofs, probabilistic or analytic, so the model provides a testbed for a ``compare and contrast" discussion of techniques. This article provides an overview of results proved in the sequence of similarly-titled articles I, III, IV and related articles. We mostly do not repeat proofs given elsewhere: instead we seek to paint a ``Big Picture" via graphics and heuristics, and emphasize open problems. Our discussion is centered around three categories of results. (i) There is a CLT for leaf heights, and the analytic proofs can be extended to provide surprisingly precise analysis of other height-related aspects. (ii) There is an explicit description of the limit {\em fringe distribution} relative to a random leaf, whose graphical representation is essentially the format of the cladogram representation of biological phylogenies. (iii) There is a canonical embedding of the discrete model into a continuous-time model, that is a random tree CTCS(n) on $n$ leaves with real-valued edge lengths, and this model turns out more convenient to study. The family (CTCS(n), n \ge 2) is consistent under a ``delete random leaf and prune" operation. That leads to an explicit inductive construction of (CTCS(n), n \ge 2) as $n$ increases, and then to a limit structure CTCS($\infty$) formalized via exchangeable partitions. Many open problems remain, in particular to elucidate a relation between CTCS($\infty$) and the $β(2,1)$ coalescent.
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