Bernhard Gittenberger, Veronika Kraus
We investigate the profile of random Pólya trees of size $n$ when only nodes of degree $d$ are counted in each level. It is shown that, as in the case where all nodes contribute to the profile, the suitably normalized profile process converges weakly to a Brownian excursion local time. Moreover, we investigate the joint distribution of the number of nodes of degree $d_1$ and $d_2$ in the levels of the tree.
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We study the local profiles of trees. We show that, in contrast with the situation for general graphs, the limit set of k-profiles of trees is convex. We initiate a study of the defining inequalities of this convex set. Many challenging problems remain open.
Random increasing k-trees represent an interesting, useful class of strongly dependent graphs for which analytic-combinatorial tools can be successfully applied. We study in this paper a notion called connectivity-profile and derive asymptotic estimates for it; some interesting consequences will also be given.
The aim of this paper is to provide an affirmative answer to a recent question by Bubeck and Linial on the local profile of trees. For a tree $T$, let $p^{(k)}_1(T)$ be the proportion of paths among all $k$-vertex subtrees (induced connected subgraphs) of $T$, and let $p^{(k)}_2(T)$ be the proportion of stars. Our main theorem states: if $p^{(k)}_1(T_n) \to 0$ for a sequence of trees $T_1,T_2,\ldots$ whose size tends to infinity, then $p^{(k)}_2(T_n) \to 1$. Both are also shown to be equivalent to the statement that the number of $k$-vertex subtrees grows superlinearly and the statement that t…
We further the study of local profiles of trees. Bubeck and Linial showed that the set of 5-profiles contains a certain polytope, namely the convex hull of d-millipedes, and they proved that the segment [0-millipede, 1-millipede] corresponds to a face of the set of 5-profiles. Our main result shows that the segment [1-millipede, 2-millipede] also corresponds to a face. Surprisingly we also show that for d > 3 the segment [d-millipede, (d+1)-millipede] is not a face of the set of 5-profiles. We do so by exhibiting new trees which are generalized millipedes with intriguing patterns for their deg…
Separation profiles of free products
通过Dvorak-Norin定理证明分离轮廓与树宽轮廓渐近等价，计算树度图的分离轮廓，拓展了自由积群的Cayley图研究。
Limit-closed Profiles
本文针对无限分离系统提出了一个缠结树定理，其特点是不对分离系统本身施加限制，而是对要在树中排列的缠结施加限制。
Rooted phylogenetic networks provide an explicit representation of the evolutionary history of a set $X$ of sampled species. In contrast to phylogenetic trees which show only speciation events, networks can also accommodate reticulate processes (for example, hybrid evolution, endosymbiosis, and lateral gene transfer). A major goal in systematic biology is to infer evolutionary relationships, and while phylogenetic trees can be uniquely determined from various simple combinatorial data on $X$, for networks the reconstruction question is much more subtle. Here we ask when can a network be unique…
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…
The vertical profile of embedded trees
本文研究节点嵌入整数轴上的有根树的垂直剖面，给出具有给定横坐标节点数的二叉树及其他树族的精确计数公式，并通过双射证明进一步细化结果。
We consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the second proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet-Melou and Janson saying that the vertical profile of a randomly labelled conditioned Galton-Watson tree converges in distribution, after suitable normalization, to th…
We prove an asymptotic Edgeworth expansion for the profiles of certain random trees including binary search trees, random recursive trees and plane-oriented random trees, as the size of the tree goes to infinity. All these models can be seen as special cases of the one-split branching random walk for which we also provide an Edgeworth expansion. These expansions lead to new results on mode, width and occupation numbers of the trees, settling several open problems raised in Devroye and Hwang [Ann. Appl. Probab. 16(2): 886--918, 2006], Fuchs, Hwang and Neininger [Algorithmica, 46 (3--4): 367--40…
Critical scaling profile for trees and connected subgraphs on the complete graph
通过分析完全图上树和连通子图的生成函数，在临界窗口内识别出两者共有的单一标度轮廓，并推测该轮廓适用于高维离散环面（d≥8）上的格点树和格点动物。
Demange (2012) generalized the classical single-crossing property to the intermediate property on median graphs and proved that the representative voter theorem still holds for this more general framework. We complement her result with proving that the linear orders of any profile which is intermediate on a median graph form a Condorcet domain. We prove that for any median graph there exists a profile that is intermediate with respect to that graph and that one may need at least as many alternatives as vertices to construct such a profile. We provide a polynomial-time algorithm to recognize wh…
Growing Binary Trees
提出一种包含生长和灭绝规则的离散演化框架，通过标签重复与分支终止机制，建立动态过程与经典无标号二叉树的直接联系，并利用结构洞察开发出高效均匀随机采样器。
It has been known for a few years that the occupation measure of several models of embedded trees converges, after a suitable normalization, to the random measure called ISE (Integrated SuperBrownian Excursion). Here, we prove a local version of this result: ISE has a (random) Hölder continuous density, and the vertical profile of embedded trees converges to this density, at least for some such trees. As a consequence, we derive a formula for the distribution of the density of ISE at a given point. This follows from earlier results by Bousquet-Mélou on convergence of the vertical profile at a …
Profile and neighbourhood complexity of graphs excluding a minor and tree-structured graphs
研究改进了排除小图和树结构图的r-轮廓复杂度和邻域复杂度的界限，并揭示了图的结构属性和算法效率。
We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root singularity in their generating function. The scaling limit involves continuous trees with branching points of order up to k+1. We derive explicit integral representations for the average profile of this k-th order multicritical continuous random tree, as well as for its history distributions measuring multi-point correlations. The latter distributions involve non-p…
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