首页/论文/math/0404126

arXiv math/04041262004-04-06math.COmath.PR

简单生成树、B-级数和Wigner过程

Simply Generated Trees, B-series and Wigner Processes

Christian Mazza

备注：: 19 pages, 1 figure

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

摘要英文摘要相关论文

摘要：

我们考虑简单生成树，并研究根平面树上的乘性函数。我们证明了相关的生成函数满足微分方程或差分方程。我们的方法考虑了Butcher理论中的B-级数，生成函数被看作是广义的龙格-库塔法

英文摘要：

We consider simply generated trees and study multiplicative functions on rooted plane trees. We show that the associated generating functions satisfy differential equations or difference equations. Our approach considers B-series from Butcher's theory, the generating functions are seen as generalized Runge-Kutta methods

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

摘要英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Simply Generated Unrooted Plane Trees

We study random unrooted plane trees with $n$ vertices sampled according to the weights corresponding to the vertex-degrees. Our main result shows that if the generating series of the weights has positive radius of convergence, then this model of random trees may be approximated geometrically by a Galton--Watson tree conditioned on having a large random size. This implies that a variety of results for the well-studied planted case also hold for unrooted trees, including Gromov--Hausdorff--Prokhorov scaling limits, tail-bounds for the diameter, distributional graph limits, and limits for the ma…

2018-09-14math.COmath.PR

The distance profile of rooted and unrooted simply generated trees

It is well-known that the height profile of a critical conditioned Galton-Watson tree with finite offspring variance converges, after a suitable normalization, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Hölder continuous of…

2021-06-21math.COmath.PR

Protection numbers in simply generated trees and Pólya trees

We determine the limit of the expected value and the variance of the protection number of the root in simply generated trees, in Pólya trees, and in unlabelled non-plane binary trees, when the number of vertices tends to infinity. Moreover, we compute expectation and variance of the protection number of a randomly chosen vertex in all those tree classes. We obtain exact formulas as sum representations, where the obtained sums are rapidly converging and therefore allowing an efficient numerical computation of high accuracy. Most proofs are based on a singularity analysis of generating functions…

2019-04-06math.CO

Cost functionals for large (uniform and simply generated) random trees

Additive tree functionals allow to represent the cost of many divide-and-conquer algorithms. We give an invariance principle for such tree functionals for the Catalan model (random tree uniformly distributed among the full binary ordered trees with given number of internal nodes) and for simply generated trees (including random tree uniformly distributed among the ordered trees with given number of nodes). In the Catalan model, this relies on the natural embedding of binary trees into the Brownian excursion and then on elementary second moment computations. We recover results first given by Fi…

2016-09-08math.CO

Reducing Simply Generated Trees by Iterative Leaf Cutting

We consider a procedure to reduce simply generated trees by iteratively removing all leaves. In the context of this reduction, we study the number of vertices that are deleted after applying this procedure a fixed number of times by using an additive tree parameter model combined with a recursive characterization. Our results include asymptotic formulas for mean and variance of this quantity as well as a central limit theorem.

2018-08-01math.CO

Simply generated non-crossing partitions

We introduce and study the model of simply generated non-crossing partitions, which are, roughly speaking, chosen at random according to a sequence of weights. This framework encompasses the particular case of uniform non-crossing partitions with constraints on their block sizes. Our main tool is a bijection between non-crossing partitions and plane trees, which maps such simply generated non-crossing partitions into simply generated trees so that blocks of size $k$ are in correspondence with vertices of outdegree $k$. This allows us to obtain limit theorems concerning the block structure of s…

2015-03-31math.COmath.PR

更多相关论文收起更多论文

简单生成树中最大保护数的分布

The distribution of the maximum protection number in simply generated trees

本文研究简单生成树中顶点最大保护数的分布，针对是否允许出度为1的顶点分别给出对数量级与双对数量级的渐近结果，并改进已有结论。

2024-11-24math.CO

大型环形地图中的距离统计

Distance statistics in large toroidal maps

我们计算了一些距离依赖的通用尺度函数，其特征在于亏格1的大型地图的距离统计。特别地，我们得到明确的概率分布的最短的不可收缩回路通过一个随机点的地图，并为两个随机点之间的距离的长度。我们的研究结果是在二分环形四边形的背景下，使用他们的编码以及标记的1-树，这是一个单一的脸和适当的整数顶点标签的属一地图。在这个框架内，上面的分布可以简单地作为良好标记的1-树的适当生成函数的缩放极限来获得，所有这些都可以用少量良好标记的平面树的基本缩放函数来表示。

2010-06-30math-phmath.COmath.MP