Y. A. Malyshkin, M. E. Zhukovskii
We prove the monadic second order 0-1 law for two recursive tree models: uniform attachment tree and preferential attachment tree. We also show that the first order 0-1 law does not hold for non-tree uniform attachment models.
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Random recursive trees: A boundary theory approach
我们表明，递归树序列的算法建设导致一个直接的证据，在相关的Doob-Martin紧化的随机递归树的收敛性，它也给出了一个表示的限制，在输入序列的算法。我们进一步表明，这种方法可以用来获得强极限定理的各种树泛函，如路径长度或维纳指数。
On tail bounds for random recursive trees
我们认为一个多变量的分布递归的总和型算法和随机树的概率分析中所产生的。我们证明了一个上尾界的解决方案，使用Beschoff的定界技术估计的拉普拉斯变换。通过随机支配将该问题追溯到二叉搜索树的相应问题。将所得结果应用于带权边的随机b元递归树和随机线性递归树的内部路径长度和Wiener指标。最后给出了这些树的Wiener指标的尾下界。
Limit Laws for Functions of Fringe trees for Binary Search Trees and Recursive Trees
证明了二叉搜索树和随机递归树的子树的函数和的极限定理。特别是，我们给出了简单的新的证据的事实，即边缘树的大小$ k=k_n $在二叉搜索树和随机递归树（总大小$ n $）的数量渐近泊松分布，如果$ k\rightarrow\infty $，和分布是渐近正态的$ k=o(\sqrt{n}) $。此外，我们证明了类似的结果的大小$ k $的子树的数量与一些所需的属性$ P $，例如，一定的固定子树$ T $的副本的数量。使用Cramér-Wold设备，我们还表明，这些随机数为不同的固定子树共同收敛到一个多元正态分布。作为一般结果的应用，我们得到了二叉搜索树或随机递归树中$\ell$ -保护结点数的一个正规极限律。证明使用一个新版本的代表Devroye，和斯坦的方法（正常和泊松近似）连同一定的耦合。
Does freezing impede the growth of random recursive trees?
研究带冻结的均匀附着模型，证明移除附着步骤可增加期望高度，而冻结不能显著降低随机递归树高度，方法基于耦合论证。
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of weights, the first order for the height of such trees has been recently established in [Electron. J. Probab. 26 (2021), Paper No. 80] by one of the authors. In this paper, we obtain the second and third orders in the asymptotic expansion of the height of weighted recursive trees, under similar assumptions. Our methods are inspired from those used to prove similar r…
Dynamics and Endogeny for recursive processes on trees
我们考虑由具有简单递归结构的无限二叉树的顶点索引的随机过程。任意一个顶点的值是两个子顶点的值的固定函数，加上一些独立的新息。内生性意味着创新正在产生。当内生性不成立时，就存在这样的动态：创新被固定，而树的边界上的一些额外的随机性被扰动。
Cut-off method for endogeny of recursive tree processes
给定一个递归分布方程的解，一个自然的（和非平凡的）问题是相应的递归树过程是否是内生的。也就是说，随机环境是否几乎必然定义树过程。我们提出了一种新的方法证明内生，适用于各种过程。作为明确的例子，我们建立内生的随机度量的非枢轴层次图定义的乘法级联和平均场最优化问题的平均场匹配和流动推销员问题在伪维数q>1。
On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees
本文考虑了随机递归树上的Bernoulli键渗流问题，树的大小为$n\gg 1$，其中对于某些固定的$c>0$，树的超临界参数为$p_n=1-c/\ln n$.已知的是，以高概率，然后存在一个独特的巨型集群的大小$G_n\sim \e^{-c}$，它遵循Schweinsberg \cite{Sch}的最近的结果，$G_n$有非高斯波动。我们提供了一个解释，这通过分析渗透的递归树的生长的不同阶段的效果。这种替代方法可能是有用的研究渗透其他类的树，例如正规树。
Percolation on random recursive trees
本文研究了随机递归树上的Bernoulli键渗流问题，当$n$趋于无穷大时，渗流参数$p(n)$收敛到$1$。渗流簇的大小自然地存储在树中。我们证明了这棵树的依分布收敛的谱系树的连续状态分支过程在离散时间。作为推论，我们获得了最大和次大渗流团簇的渐近尺寸，从而扩展了Bertoin（2014）最近的工作，该工作涉及超临界状态下的团簇尺寸。在第二部分中，我们表明，同样的极限树出现在研究的树组件出现从一个连续时间的随机递归树的破坏。我们评论的连接，我们的第一个结果伯努利键渗流。
A random recursive tree model with doubling events
提出一种新的随机树模型，在随机递归树基础上引入全局加倍事件，研究其大小、度分布、高度轮廓的渐近性质，并给出高度下界。
Infinite collisions of simple random walks on random recursive trees generated by Bernoulli sequences
研究伯努利序列生成的随机递归树，通过从特定图递归构建，证明其几乎必然恰有一个拓扑端，且具有无限碰撞性质，即两个独立简单随机游走几乎必然无限次碰撞。
On a sufficient condition for explosion in CMJ branching processes and applications to recursive trees
本文通过有限时间内产生无限路径提供了 CMJ 分支过程爆炸的充分条件，并应用于带适应度的递归树模型，揭示了无限树中节点度数从无限到局部有限的相变现象。
A Necessary and Sufficient Condition for the Tail-Triviality of a Recursive Tree Process
给定一个递归分布方程（RDE）及其解$μ$，我们考虑树索引不变过程称为具有边际$μ$的递归树过程（RTP）.本文引入了一种不同于Aldous和Bandyopadhyay（2005）定义的二元唯一性性质，并证明了该性质等价于RTP的尾平凡性，从而得到了判定RTP的尾平凡性的一个充要条件.作为应用，我们考虑了Aldous（2000）在无限正则树上构造的冻结渗流过程，并证明了相关的RTP具有平凡尾。
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen at random with probability proportional to its weight. In the case where the total weight of the tree at step $n$ grows polynomially in $n$, we obtained in (Pain-Sénizergues 2022) an asymptotic expansion for the height of the tree, which falls into the university class of the maximum of branching random walks. In this paper, we consider the case of a total weight growing sub-polynomially in $n$ and obtain asymptotics for the height of the tree …
Atypical Decay Rates for Atypical Heights in Random Recursive Trees
本文建立了随机递归树高度的大偏差概率，发现上尾呈多项式衰减，下尾呈拉伸指数衰减，且下尾具有异常前因子，其增长速度慢于任何n重迭代对数。
We consider a random process on recursive trees, with three types of events. Vertices give birth at a constant rate (growth), each edge may be removed independently (fragmentation of the tree) and clusters (or trees) are frozen with a rate proportional to their sizes (isolation of connected component). A phase transition occurs when the isolation is able to stop the growth fragmentation process and cause extinction. When the process survives, the number of clusters increases exponentially and we prove that the normalized empirical measure of clusters a.s. converges to a limit law on recursive …
We study two models of growing recursive trees. For both models, initially the tree only contains one vertex $u_1$ and at each time $n\geq 2$ a new vertex $u_n$ is added to the tree and its parent is chosen randomly according to some rule. In the \emph{weighted recursive tree}, we choose the parent $u_k$ of $u_n$ among $\{u_1,u_2,\dots, u_{n-1}\}$ with probability proportional to $w_k$, where $(w_n)_{n\geq1}$ is some deterministic sequence that we fix beforehand. In the \emph{affine preferential attachment tree with fitnesses}, the probability of choosing any $u_k$ is proportional to $a_k+\mat…
Hoppe trees, random recursive sets and their barycentre
我们考虑一个递归定义的随机点集及其重心，其中随机集由以下归纳规则构造：给定一个$n-1$点的实现，其中之一是随机挑选的，并作为源的$n$ -th点。讨论了这一随机集重心的渐近性质。主要的分析依赖于Hoppe树的分析，我们推导出了总长度和Wiener指标的联合分布的极限定理。
On the asymptotic behaviour of random recursive trees in random environment
我们考虑在随机环境中生长随机递归树，其中在每一步中，根据概率分布将一个新的顶点（通过随机长度的边）附加到现有的树顶点，该概率分布分配树顶点的质量与它们的随机权重成比例。本文的主要目的是研究新插入的顶点到树根的距离的渐近性态以及当步数趋于无穷大时，平均出顶点数的渐近性态。大多数结果是在假设随机权重具有独立同分布因子的乘积形式下得到的。
Degree distribution in the lower levels of the uniform recursive tree
本文考虑了均匀随机递归树在$n$步后的第$k$层，证明了对于每一个$t\in(0,1)$，度大于$t\log n$的结点的比例几乎必然收敛到$(1-t)^k$，即$n\to\infty$.此外，我们证明了度$d$节点的数量在第一级是渐近泊松分布的平均值为1，而且，他们是渐近独立的$d=1,2,...$。
Recursive construction of continuum random trees
我们介绍了一个一般的递归方法来构建连续随机树（CRT）的随机串珠，即，任何随机间隔配备了一个随机离散的概率测度的独立副本，并从相关的结构。我们证明了这些CRT的存在作为一个新的应用程序的不动点方法的递归分布方程的形式化在高的一般性由Aldous和Bandyopadhyay。我们应用这种递归方法显示各种树生长过程的CRT的收敛。我们注意到现有的自相似CRT的替代结构的意义上的哈斯，Miermont和斯蒂芬森，我们给随机紧凑的R-树，描述的家谱Bertoin的自相似增长片段的第一次建设。在即将到来的工作中，我们开发进一步的应用嵌入问题的CRT，提供了一个二进制嵌入的稳定的线打破建设，解决了一个开放的问题Goldschallow和哈斯。
Cutting edges at random in large recursive trees
我们评论旧的和新的结果有关的随机递归树（RRT）的破坏，其中它的边缘被切断一个接一个在一个统一的随机顺序。特别是，我们研究了当树的大小趋于无穷大时，隔离或断开某些特殊顶点所需的步骤数。新的概率解释给出了所谓的割树和树的组件的大小，这两个编码不同方面的销毁过程。最后，我们建立了连接到伯努利键渗流大RRT的，并提出最近的结果在超临界状态下的集群大小。
We study the broadcasting problem when the underlying tree is a random recursive tree. The root of the tree has a random bit value assigned. Every other vertex has the same bit value as its parent with probability $1-q$ and the opposite value with probability $q$, where $q \in [0,1]$. The broadcasting problem consists in estimating the value of the root bit upon observing the unlabeled tree, together with the bit value associated with every vertex. In a more difficult version of the problem, the unlabeled tree is observed but only the bit values of the leaves are observed. When the underlying …
Interacting particle systems can often be constructed from a graphical representation, by applying local maps at the times of associated Poisson processes. This leads to a natural coupling of systems started in different initial states. We consider interacting particle systems on the complete graph in the mean-field limit, i.e., as the number of vertices tends to infinity. We are not only interested in the mean-field limit of a single process, but mainly in how several coupled processes behave in the limit. This turns out to be closely related to recursive tree processes as studied by Aldous a…
Subcritical percolation and network archaeology on random recursive tree substrate networks
研究随机递归树基底网络的网络考古学问题，利用辅助亚临界键渗流揭示树状重整化结构，通过在最大辅助渗流组件内应用乔丹中心性，为循环网络构建确定性大小的根置信集。
We consider random recursive trees that are grown via community modulated schemes that involve random attachment or degree based attachment. The aim of this paper is to derive general techniques based on continuous time embedding to study such models. The associated continuous time embeddings are not branching processes: individual reproductive rates at each time depend on the composition of the entire population at that time. Using stochastic analytic techniques we show that various key macroscopic statistics of the continuous time embedding stabilize, allowing asymptotics for a host of funct…
A degree-biased cutting process for random recursive trees
本文提出了一种基于度数偏好的随机递归树切割过程，推导了切割次数的递归公式，并证明其收敛性与稳定分布的关系。
Consider a Crump-Mode-Jagers process generated by an increasing random walk whose increments have finite second moment. Let $Y_k(t)$ be the number of individuals in generation $k\in \mathbb N$ born in the time interval $[0,t]$. We prove a law of the iterated logarithm for $Y_k(t)$ with fixed $k$, as $t\to +\infty$. As a consequence, we derive a law of the iterated logarithm for the number of vertices at a fixed level $k$ in a random recursive tree, as the number of vertices goes to $\infty$.
Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We prove a functional limit theorem for the vector-valued process $(X_{[n^t]}(1),\ldots, X_{[n^t]}(k))_{t\geq 0}$, for each $k\in\mathbb N$. We show that after proper centering and normalization, this process converges weakly to a vector-valued Gaussian process whose components are integrated Brownian motions. This result is deduced from a functional limit theorem for Crump-Mode-Jagers branching processes generated by increasing random walks with increments that have finite second moment.
Fires on large recursive trees
我们考虑了具有$n$顶点的均匀随机递归树上的随机动力学。接着，以均匀的随机顺序，每个边缘要么以某种概率$p_n$着火，要么以概率$1-p_n$防火。火在树上蔓延，只能被防火边缘阻止。我们首先考虑燃烧顶点和耐火顶点的比例为$n\to\infty$，并证明了当$p_n$为$\ln n/n$阶时的相变。然后，我们研究了防火森林的连通性，更准确地说，一个巨大的组件的存在。最后，我们研究了烧毁子树的大小。
On asymptotics of two non-uniform recursive tree models
本文研究了两类非均匀随机递归树的性质。在第一个模型中，权重被分配给每个节点，从而改变连接概率。我们将这些树称为加权递归树。在第二个模型中，在对称群上选择了不同的分布而不是均匀分布，即一个随机洗牌分布。这些树被称为有偏递归树。研究了这两种模型的分枝数、叶数、节点深度等性质。重点是渐近结果和比较均匀随机递归树。结果表明，当节点数增加时，加权递归树的研究性质在许多情况下接近于均匀递归树。相反，有偏递归树表现出不同的行为，但方法均匀递归树的参数的riffle洗牌分布。
A weighted recursive tree is an evolving tree in which vertices are assigned random vertex-weights and new vertices connect to a predecessor with a probability proportional to its weight. Here, we study the maximum degree and near-maximum degrees in weighted recursive trees when the vertex-weights are almost surely bounded and their distribution function satisfies a mild regularity condition near zero. We are able to specify higher-order corrections to the first order growth of the maximum degree established in prior work. The accuracy of the results depends on the behaviour of the weight dist…
In this paper, several properties of plain-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and variance of the degree variable via a \Polya\ urn approach. In addition, we look into a topological index, the Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index by using recurrence methods. We also provide several evidence in favor of our conjecture that the Zagreb index of P…
We provide a new characterisation of Duquesne and Le Gall's $α$-stable tree, $α\in(1,2]$, as the solution of a recursive distribution equation (RDE) of the form $\mathcal{T}\overset{d}{=}g(ξ,\mathcal{T}_i, i\geq0)$, where $g$ is a concatenation operator, $ξ= (ξ_i, i\geq 0)$ a sequence of scaling factors, $\mathcal{T}_i$, $i \geq 0$, and $\mathcal{T}$ are i.i.d. trees independent of $ξ$. This generalises a version of the well-known characterisation of the Brownian Continuum Random Tree due to Aldous, Albenque and Goldschmidt. By relating to previous results on a rather different class of RDE, w…
Random recursive trees and preferential attachment trees are random split trees
我们考虑了线性偏好连接树，并证明了它们可以被看作是Devroye（1999）意义下的随机分裂树，尽管具有无限的潜在分支。特别地，这适用于随机递归树和标准优先连接树。一个应用程序的总和在所有对节点的共同数目的祖先。
Distances in $m$-ary recursive trees
本文引入$m$叉递归树并分析其距离，推导深度分布，通过耦合得到高度极限，并用加权范数收缩法建立维纳指数与内部路径长度的二元极限定律。
The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes
我们考虑了一个无限递归树的自然销毁过程，删除每个边后，一个独立的指数时间。直到时间t的破坏由N的划分$Π$（t）编码为连接顶点的块。尽管缺乏交换性，就像可交换碎裂过程一样，过程$Π$是马尔可夫的，其跃迁由分裂率测度r决定。然而，有点令人惊讶的是，r未能满足通常的可交换碎片的位错措施的可积性条件。我们进一步观察到，依赖于时间的归一化使我们能够定义$Π$（t）的块的权重。我们研究了这些权重的过程，并指出了与Ornstein-Uhlenbeck型过程的联系。
Size distribution of clusters in site-percolation on random recursive tree
本文严格证明了随机递归树上位点渗流中团簇大小的幂律分布、最大团簇阶数及其标度极限，基于多类型分支过程嵌入和键渗流耦合。
We examine a discrete random recursive tree growth process that, at each time step, either adds or deletes a node from the tree with probability $p$ and $1-p$, respectively. Node addition follows the usual uniform attachment model. For node removal, we identify a class of deletion rules guaranteeing the current tree $T_n$ conditioned on its size is uniformly distributed over its range. By using generating function theory and singularity analysis, we obtain asymptotic estimates for the expectation and variance of the tree size of $T_n$ as well as its expected leaf count and root degree. In all …
We identify the mean growth of the independence number of random binary search trees and random recursive trees and show normal fluctuations around their means. Similarly we also show normal limit laws for the domination number and variations of it for these two cases of random tree models. Our results are an application of a recent general theorem of Holmgren and Janson on fringe trees in these two random tree models.
The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform permutations, biased riffle shuffles. The statistics that are treated include the number of nodes with a given number of descendants, the depth of the tree, and the number of branches. The model yields the uniform recursive trees as a certain limit, some new results for the uniform case are obtained as well.
In this paper, we study the joint behaviour of the degree, depth and label of and graph distance between high-degree vertices in the random recursive tree. We generalise the results obtained by Eslava and extend these to include the labels of and graph distance between high-degree vertices. The analysis of both these two properties of high-degree vertices is novel, in particular in relation to the behaviour of the depth of such vertices. In passing, we also obtain results for the joint behaviour of the degree and depth of and graph distance between any fixed number of vertices with a prescribe…
Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We are interested in the asymptotic behavior of $X_n(k)$ for intermediate levels $k=k_n$ satisfying $k_n\to\infty$ and $k_n=o(\log n)$ as $n\to\infty$. In particular, we prove weak convergence of finite-dimensional distributions for the process $(X_n ([k_n u]))_{u>0}$, properly normalized and centered, as $n\to\infty$. The limit is a centered Gaussian process with covariance $(u,v)\mapsto (u+v)^{-1}$. One-dimensional distributional convergence of $X_n(k_n)$, properly normalized and centered, was…
A study of centrality measures in random recursive trees
本文研究了随机递归树中五种经典中心性度量在时间到达顺序和根节点相对中心性上的行为，分析了根节点成为最中心节点的概率及其预期排名，以及中心性顶点的出生时间分布。
We propose a method for cutting down a random recursive tree that focuses on its higher degree vertices. Enumerate the vertices of a random recursive tree of size $n$ according to a decreasing order of their degrees; namely, let $(v^{(i)})_{i=1}^{n}$ be so that $deg(v^{(1)}) \geq \cdots \geq deg (v^{(n)})$. The targeted, vertex-cutting process is performed by sequentially removing vertices $v^{(1)}$, $v^{(2)}, \ldots, v^{(n)}$ and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed. The total number of steps for this procedu…
Plane recursive trees, Stirling permutations and an urn model
我们利用平面递归树和斯特林置换之间的双射，这产生了等价的一些结果先前证明分别由不同的方法为两种类型的对象，以及一些新的结果。我们还证明了随机Stirling排列中上升、下降和平稳数的联合分布的结果。证明使用了一个有趣的广义波利亚瓮
Can the root cluster remain largest forever in random recursive tree percolation?
研究随机递归树上的伯努利键渗流，证明根簇以严格正概率永远保持最大簇，并分析相关概率随保留参数p的变化性质。
Recursive Paintboxes and the Martin Boundary of the Hoffman Rooted-Tree Graph
本文确定了Hoffman有根树图的Martin边界，证明其由递归油漆盒参数化，并给出了递归Ewens族的极值分解。
Parking on the Random Recursive Tree
本文研究随机递归树上的停车过程，证明相变位于密度0，并给出正车流量高概率出现的临界窗口及二元到达情形下的密度阈值。
The generalized Zagreb index for non-plane and plane recursive trees
本文研究非平面与平面递归树的广义Zagreb指数，采用矩转移方法证明其矩的一阶渐近性及极限定律，发现非平面树极限为正态分布，平面树三次和四次幂为非正态分布。
A uniform recursive tree on $n$ vertices is a random tree where each possible $(n-1)!$ labeled recursive rooted tree is selected with equal probability. In this paper we introduce and study weighted trees, a non-uniform recursive tree model departing from the recently introduced Hoppe trees. This class generalizes both uniform recursive trees and Hoppe trees. The generalization provides diversity among the nodes, making the model more flexible for applications. We also analyze the number of leaves, the height, the depth, the number of branches, and the size of the largest branch in these weigh…
A Random Walk Approach to Broadcasting on Random Recursive Trees
本文通过随机游走与记忆效应的联系，提出新方法分析随机递归树上多数估计器，并扩展至递增树与形状可交换树，推广了先前研究。
We introduce a non-increasing tree growth process $((T_n,σ_n),\, n\ge 1)$, where $T_n$ is a rooted labeled tree on $n$ vertices and $σ_n$ is a permutation of the vertex labels. The construction of $(T_{n},σ_n)$ from $(T_{n-1},σ_{n-1})$ involves rewiring a random (possibly empty) subset of edges in $T_{n-1}$ towards the newly added vertex; as a consequence $T_{n-1} \not\subset T_n$ with positive probability. The key feature of the process is that the shape of $T_n$ has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotonous in the pro…
Multiple isolation of nodes in recursive trees
我们介绍了随机递归树中的几个节点通过连续删除随机边隔离的问题，并研究了隔离所需的随机切割的数量。特别地，我们分析了对于三种不同的选择规则，在大小为n的随机递归树中隔离$\ell$选择的节点所需的随机切割的数量，即（i）隔离标记为1，2，.的所有节点，$\ell$（因此节点位于树的根附近），（ii）隔离标记为n+1- $\ell$、n+2- $\ell$、.的所有节点，n（因此节点位于树的边缘），以及（iii）隔离树中的$\ell$节点，这些节点在开始边缘去除过程之前被随机选择。使用生成函数的方法，我们确定这些选择规则的限制分布行为的切割隔离所有选定的节点，为$\ell$固定和n趋于无穷大。
Depth properties of scaled attachment random recursive trees
本文研究了一类随机递归树的深度性质，其中每个结点i都与随机结点iX_i相连，X_0，.，X_n是独立同分布的序列。取值于[0，1）的随机变量。我们称这种树为标度连接随机递归树（SART）。我们证明了典型深度D_n、最大深度（或高度）H_n和SARRT的最小深度M_n由D_n \sim μ ^{-1} \log n、H_n \sim α_{\max} \log n和M_n \sim α_{\min} \log n渐近给出，其中μ，当X_0有密度时，α_{\max}和α_{\min}是只与X_0的分布有关的常数。特别地，给出了不使用分支随机游动的均匀随机递归树H_n \sim e \log n的高度的一个新的初等证明.
A recursive function on a tree is a function in which each leaf has a given value, and each internal node has a value equal to a function of the number of children, the values of the children, and possibly an explicitly specified random element $U$. The value of the root is the key quantity of interest in general. In this first study, all node values and function values are in a finite set $S$. In this note, we describe the limit behavior when the leaf values are drawn independently from a fixed distribution on $S$, and the tree $T_n$ is a random Galton--Watson tree of size $n$.
Broadcasting induced colourings of random recursive trees and preferential attachment trees
本文针对随机线性优先连接树的广播诱导二着色问题，结合Pólya瓮模型与解析组合学方法，证明了多类相关统计量经重标度后的极限分布，拓展了随机优先连接树的键渗流研究。
We review fundamentals underlying binary search trees and digital search trees, with (atypical) emphasis on recursive formulas for associated probability generating functions. Other topics include higher moments of BST search costs and combinatorics for a certain finite-key analog of DSTs.
The rank and layer distributions in random recursive trees
本文推导随机递归树中秩与层分布的精确递推式，证明二者分别呈阶乘、几何衰减且对应节点数自平均，提出不依赖标签或根节点的通用分析框架。
The dual tree of a recursive triangulation of the disk
在圆盘的递归叠层中，人们试图随机地一个接一个地添加和弦;如果一个和弦不与任何先前插入的和弦相交，则保留并插入该和弦。Curien和Le Gall [Ann. Probab. 39（2011）2224-2270]已经证明了弦的集合收敛到由连续过程$\mathscr{M}$编码的盘的极限三角测量。基于一种新的方法类似的想法，从所谓的收缩方法在函数空间中，我们证明了，当适当的重新缩放，离散层的平面对偶收敛几乎肯定在Gromov-Hausdorff意义上的极限实树$\mathscr{T}$，这是由$\mathscr{M}$编码。这证实了Curien和Le Gall的一个猜想。
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