A proof of the rooted tree alternative conjecture
Mykhaylo Tyomkyn
Bonato和Tardif证明了与给定的树T相互嵌入的树的同构类的数目要么是1要么是无穷大。我们证明了类似的猜想为有根树。本文还讨论了局部有限树的原猜想，并给出了一些新的证明。
Bonato and Tardif conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or infinite. We prove the analogue of their conjecture for rooted trees. We also discuss the original conjecture for locally finite trees and state some new conjectures.
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On the enumeration of records of rooted trees and rooted forests
本文将有根Cayley树及有根森林记录的生成函数与Cayley树函数相关联，得到了优美的函数方程并探讨了其结果。
Let $\T_{n}$ be the set of rooted labeled trees on $\set{0,...,n}$. A maximal decreasing subtree of a rooted labeled tree is defined by the maximal subtree from the root with all edges being decreasing. In this paper, we study a new refinement $\T_{n,k}$ of $\T_n$, which is the set of rooted labeled trees whose maximal decreasing subtree has $k+1$ vertices.
On a family of relations of rooted tree maps
该研究证明了有根树映射的无穷序列关系，同时给出了有根树映射空间的一组基，推进了相关组合数学领域的研究。
We give exact formulas for the transmission (i.e. the sum of all distances between vertices) of perfect trees and rooted powers of (connected finite) graphs.
In the work [4] tree-rooted planar cubic maps with marked directed edge (not in this tree) were enumerated. The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is Catalan number. In this work a marked directed edge is not demanded, i.e. we enumerate tree-rooted planar cubic maps. Formulas are more complex, of course, but not significantly.
We obtain a differential equation for the enumeration of the path length of general increasing trees. By using differential operators and their combinatorial interpretation we give a bijective proof of a version of Faà di Bruno formula, and model the generation of ballot and Dyck paths. We get formulas for its enumeration according with the height of their lattice points. Recursive formulas for the enumeration of enriched increasing trees and forests with respect to the height of their internal and external vertices are also obtained. Finally we present a generalized form of all those results …
We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at random has a given rank. We prove that this probability converges to a limit as the tree size goes to infinity.
In a rooted tree, we call a vertex {\em balanced} if it is at equal distance from all its descendant leaves. We count balanced vertices in three different tree varieties. For decreasing binary trees, we can prove that the probability that a vertex chosen uniformly at random from the set of all trees of a given size is balanced is monotone decreasing.
Prüfer codes on vertex-colored rooted trees
本文扩展Prüfer编码至顶点着色有根树，定义规范标记与VCPC，证明其可判定树同构、子树关系及子式同构，应用于货运网络分解。
0-rotatability of classes of rooted symmetric trees. Are rooted symmetric trees 0-rotatable?
本文证明至多3层的根对称树及两类4层根对称树（腿长≤3的对称蜘蛛树和对称香蕉树）是0-可旋转的，并猜想所有蜘蛛树及对称根树均0-可旋转。
On locally finite ordered rooted trees and their rooted subtrees
本文从枚举与组合角度比较 game-over 攻击的动态多项式时间算法和暴力子树检查，推广双重指数序列结果，并刻画有限及局部有限有序有根树中低高度有根子树数量为多项式界的条件。
The perturbation expansion of the solution of a fixed point equation or of an ordinary differential equation may be expressed as a power series in the perturbation parameter. The terms in this series are indexed by rooted trees and depend on a parameter in the equation in a way determined by the structure of the tree. Power series of this form may be considered more generally; there are two interesting and useful group structures on these series, corresponding to operations of composition and substitution. The composition operation defines the Butcher group, an infinite dimensional group that …
We find a simple, closed formula for the proportion of vertices which are $k$-protected in all unlabeled rooted plane trees on $n$ vertices. We also find that, as $n$ goes to infinity, the average rank of a random vertex in a tree of size $n$ approaches 0.727649, and the average rank of the root of a tree of size $n$ approaches 1.62297.
In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted $U$-polynomial for every $k$; by this we mean that the polynomials agree on terms with degree at most $k+1$. The main tool for this construction is a generalization of the $U$-polynomial to rooted graphs, which we introduce and study in this article. Most notably we show that rooted trees can be reconstructed from its rooted $U$-polynomial.
Determining whether two graphs are isomorphic is an important and difficult problem in graph theory. One way to make progress towards this problem is by finding and studying graph invariants that distinguish large classes of graphs. Stanley conjectured that his chromatic symmetric function distinguishes all trees, which has remained unresolved. Recently, Hasebe and Tsujie introduced an analogue of Stanley's function for posets, called the strict order quasisymmetric function, and proved that it distinguishes all rooted trees. In this paper, we devise a procedure to explicitly reconstruct a roo…
Decks of rooted binary trees
该研究围绕有根二叉树的卡堆与多重卡堆展开极值问题分析，给出树重构所需最小卡堆规模的上下界，刻画了卡堆规模取极值的树，并完成了$k$-通用树的穷举计算。
Bijective counting of tree-rooted maps and shuffles of parenthesis systems
树根映射，即具有一个特殊生成树的平面映射，其大小为$n$的个数为C（n）C（n+1），其中C（n）=二项式（2n，n）/（n+1）是第n个Catalan数。我们提出了一个（期待已久的）简单的双射解释了这一结果。我们证明了我们的双射是同构于前递归结构上的括号系统由于科里，Dulucq和Viennot洗牌。
Representing distance-hereditary graphs with multi-rooted trees
本文提出树状网络这一根树的推广概念，证明距离遗传图类可由带标签树状网络解释，类似余图类由带标签根树解释，建立了距离遗传图与多根树结构的对应关系。
Seo and Shin showed that the number of rooted trees on $[n+1]$ such that the maximal decreasing subtree with the same root has $k+1$ vertices is equal to the number of functions $f:[n]\to[n]$ such that the image of $f$ contains $[k]$. We give a bijective proof of this theorem.
We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial objects, including the ordered partitions of $[n]$ all of whose left-to-right minima occur at odd locations and the drawings of rooted plane trees with $n+1$ vertices. These are sequences of rooted plane trees such that each tree in a sequence can be obtained from the next one by removing a leaf.
In this paper, we give a decomposition of triply rooted trees into three doubly rooted trees. This leads to a combinatorial interpretation of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory, and proved by Younsi by using the Hurwitz identity on multivariate Abel polynomials. We also give a bijection between the set of functions from $[n+1]$ to $[n]$ and the set of triply rooted trees on $[n]$, which leads to the refined enumeration of functions from $[n+1]$ to $[n]$ with respect to the number of elements in the orbit of $n+1$ and the number of period…
We consider planar cubic maps, i.e. connected cubic graphs imbedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number.
In this paper we enumerate the cardinalities for the set of all vertices of outdegree $\ge k$ at level $\ge \ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature.
Rooted triples, rooted binary phylogenetic trees on three leaves, are sufficient to encode rooted binary phylogenetic trees. That is, if $\mathcal T$ and $\mathcal T'$ are rooted binary phylogenetic $X$-trees that infers the same set of rooted triples, then $\mathcal T$ and $\mathcal T'$ are isomorphic. However, in general, this sufficiency does not extend to rooted binary phylogenetic networks. In this paper, we show that trinets, phylogenetic network analogues of rooted triples, are sufficient to encode rooted binary orchard networks. Rooted binary orchard networks naturally generalise roote…
Binary rooted trees, both in the ordered and in the un-ordered case, are well studied structures in the field of combinatorics. The aim of this work is to study particular patterns in these classes of trees. We consider completely unbalanced subtrees, where unbalancing is measured according to the so-called Colless's index. The size of the biggest unbalanced subtree becomes then a new parameter with respect to which we find several enumerations.
The independence polynomial of a graph $G$ is \[I(G,x)=\sum\limits_{k\ge 0}i_k(G)x^k,\] where $i_k(G)$ denotes the number of independent sets of $G$ of size $k$ (note that $i_0(G)=1$). In this paper we show a new method to prove real-rootedness of the independence polynomials of certain families of trees. In particular we will give a new proof of the real-rootedness of the independence polynomials of centipedes (Zhu's theorem), caterpillars (Wang and Zhu's theorem), and we will prove a conjecture of Galvin and Hilyard about the real-rootedness of the independence polynomial of the so-called Fi…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on $\{1,2,\ldots,n\}$. Classical formulas for counting various trees such as rooted trees, bipartite trees, tripartite trees, plane trees, $k$-ary plane trees, $k$-edge colored trees follow immediately from our recursive relations.
Free hyperplane arrangements associated to labeled rooted trees
每个标记的有根树与超平面排列相关联，该超平面排列不受由该树的顶点的深度给出的指数的影响。这些安排的交叉格描述通过偏序集的森林。这些偏序集被用来定义余代数，其对偶代数被证明有一个简单的表示的生成元和关系。
Let $\mathcal{T}_n$ be the set of trees with $n$ vertices. Suppose that each tree in $\mathcal{T}_n$ is equally likely. We show that the number of different rooted trees of a tree equals $(μ_r+o(1))n$ for almost every tree of $\mathcal{T}_n$, where $μ_r$ is a constant. As an application, we show that the number of any given pattern in $\mathcal{T}_n$ is also asymptotically normally distributed with mean $\sim μ_M n$ and variance $\sim σ_M n$, where $μ_M, σ_M$ are some constants related to the given pattern. This solves an open question claimed in Kok's thesis.
The interior and exterior activities of bases of a matroid are well-known notions that for instance permit one to define the Tutte polynomial. Recently, we have discovered correspondences between the regions of gainic hyperplane arrangements and coloredlabeled rooted trees. Here we define a general activity theory that applies in particular to no-broken circuit (NBC) sets and labeled colored trees. The special case of activity \textsf{0} was our motivating case. As a consequence, in a gainic hyperplane arrangement the number of bounded regions is equal to the number of the corresponding colore…
Assigned rational functions of a rooted tree
该研究定义有根树顶点的指派有理函数，推导邻接与拉普拉斯矩阵特征多项式的有理函数乘积表达式，应用于平衡树并重点研究Bethe树，还构建了保谱的树合并方法，改进了有根树谱性质的现有研究结果。
Two decades ago, Chauve, Dulucq and Guibert showed that the number of rooted trees on the vertex set $[n+1]$ in which exactly $k$ children of the root are lower-numbered than the root is $\binom{n}{k} \, n^{n-k}$. Here I give a simpler proof of this result.
Coalgebras, bialgebras and Rota-Baxter algebras from shuffles of rooted forests
本文将词洗牌性质推广到根树与根森林，构造根树上的余积和积以建立洗牌双代数及其对偶双代数，刻画并计数原始树，并利用修正洗牌证明Rota-Baxter代数中的性质。
Enumeration of multi-rooted plane trees
本文给出了多根平面树计数的闭式表达式、递推关系及生成函数显式形式，并附带新二项式恒等式和超几何函数猜想。
Universal rooted phylogenetic tree shapes and universal tanglegrams
本文研究有根二叉树和缠结图的通用结构规模，给出上下界并计算 n≤11 的最小通用树，部分结果推广到 d 叉情形。
We provide formulas for generating functions of many types of paths in various rooted tree structures. We compute the $k$th moment of the generating functions for various types of vertical paths. In two specific familes of trees we find exact closed formulas for expectations and their asymptotic values. Some of these closed formulas are surprisingly simple.
Cyclic sieving phenomena for trees and tree-rooted maps
本文证明了角根平面树（有序树）在四种循环群作用下的循环筛分现象，并推广到树根平面地图。
In this paper we enumerate and give bijections for the following four sets of vertices among rooted ordered trees of a fixed size: (i) first-children of degree $k$ at level $\ell$, (ii) non-first-children of degree $k$ at level $\ell-1$, (iii) leaves having $k-1$ elder siblings at level $\ell$, and (iv) non-leaves of outdegree $k$ at level $\ell-1$. Our results unite and generalize several previous works in the literature.
We investigate a trivariate polynomial associated with rooted trees. It generalises a bivariate polynomial for rooted trees that was recently introduced by Liu. We show that this polynomial satisfies a deletion-contraction recursion and can be expressed as a sum over maximal antichains. Several combinatorial quantities can be obtained as special values, in particular the number of antichains, maximal antichains and cutsets. We prove that two of the three possible bivariate specialisations characterise trees uniquely up to isomorphism. One of these has already been established by Liu, the other…
Given a rooted tree $T$ with leaves $v_1,v_2,\ldots,v_n$, we define the ancestral matrix $C(T)$ of $T$ to be the $n \times n$ matrix for which the entry in the $i$-th row, $j$-th column is the level (distance from the root) of the first common ancestor of $v_i$ and $v_j$. We study properties of this matrix, in particular regarding its spectrum: we obtain several upper and lower bounds for the eigenvalues in terms of other tree parameters. We also find a combinatorial interpretation for the coefficients of the characteristic polynomial of $C(T)$, and show that for $d$-ary trees, a specific valu…
We define a bivariate polynomial for unlabeled rooted trees and show that the polynomial of an unlabeled rooted tree $T$ is the generating function of a class of subtrees of $T$. We prove that the polynomial is a complete isomorphism invariant for unlabeled rooted trees. Then, we generalize the polynomial to unlabeled unrooted trees and we show that the generalized polynomial is a complete isomorphism invariant for unlabeled unrooted trees.
Counting Rooted Trees: The Universal Law t(n) ~ C ρ^{-n} n^{-3/2}
使用标准多重集、序列、有向循环和循环构造及其限制的组合递归定义的组合类T具有正收敛半径的生成级数T（z）;对于其中的大多数，可以使用一个简单的测试来快速证明渐近的形式与有根树类的渐近形式相同：C ρ^{-n} n^{-3/2}其中ρ是T的收敛半径。
The Perron value $ρ(T)$ of a rooted tree $T$ has a central role in the study of the algebraic connectivity and characteristic set, and it can be considered a weight of spectral nature for $T$. A different, combinatorial weight notion for $T$ - the moment $μ(T)$ - emerges from the analysis of Kemeny's constant in the context of random walks on graphs. In the present work, we compare these two weight concepts showing that $μ(T)$ is "almost" an upper bound for $ρ(T)$ and the ratio $μ(T)/ρ(T)$ is unbounded but at most linear in the order of $T$. To achieve these primary goals, we introduce two new…
Positivity Rigidity for Grossman-Larson Characters and the Kingman Face of the Hoffman Rooted-Tree Graph
本文分类了 Grossman-Larson Hopf 代数在根树基上的非负实特征，揭示其由 Kingman paintbox 参数化，并刻画了 Hoffman 根树图上的中心测度面及 Martin 边界结构。
A rooted tree T is a poset whose Hasse diagram is a graph-theoretic tree having a unique minimal element. We study rowmotion on antichains and lower order ideals of T. Recently Elizalde, Roby, Plante and Sagan considered rowmotion on fences which are posets whose Hasse diagram is a path (but permitting any number of minimal elements). They showed that in this case, the orbits could be described in terms of tilings of a cylinder. They also defined a new notion called homometry which means that a statistic takes a constant value on all orbits of the same size. This is a weaker condition than the…
Based on Hopf algebra of rooted trees introduced by Connes and Kreimer, we construct a class of linear maps on noncommutative polynomial algebra in two indeterminates, namely rooted tree maps. We also prove that their maps induce a class of relations among multiple zeta values.
Let $\mathcal{D}_0:= \{x + iy \ \vert x, y >0\}$, and let $(L, R)$ be a pair of Möbius transformations corresponding to $\mathrm{SL}_2(\mathbb{N}_0)$ matrices such that $R(\mathcal{D}_0)$ and $L(\mathcal{D}_0)$ are disjoint. Given such a pair (called a left-right pair), we can construct a directed graph $\mathcal{F}(L, R)$ with vertices $\mathcal{D}_0$ and edges $\{(z, R(z))\}_{z \in \mathcal{D}_0} \cup \{(z, L(z))\}_{z \in \mathcal{D}_0}$, which is a collection of infinite binary trees. We answer two questions of Nathanson by classifying all the pairs of elements of $\mathrm{SL}_2(\mathbb{N}_…
Transversals in trees
有根树中的截线是满足从根到叶的每一条路径的任何节点集合。我们用c（T，k）表示有根树T中大小为k的断面的数目。我们定义了所有具有n个结点的有根树的集合上的一个偏序，即树T继树T'如果c（T，k）对于所有k至少为c（T'，k），并且对于至少一个k严格大于c（T '，k）.我们证明，对于每一个选择的正整数d和n，所有有根树的n个节点，每个节点有最多d个孩子的集合有一个唯一的极小元关于这个偏序，我们描述了这棵树。
We propose a novel approach for studying rooted trees by using functions that we will call descent functions. We provide a construction method for rooted trees that allows to study their properties through the use of descent functions. Moreover, in this way, we are able to compose rooted trees with each other. Such a new composition of rooted trees is a very powerful tool applied in this paper in order to obtain important results as the creation of new rational and Pythagorean trees.
Rooted trees, non-rooted trees and hamiltonian B-series
研究有根树与无根树的关系，提出无根树中替代李括号的预李乘积反称化方法。
We present a bijection from planar reduced trees to planar rooted hypertrees, which extends Knuth's rotation correspondence between planar binary trees and planar rooted trees. The operadic counterpart of the new bijection is explained. Related to this, the space of planar reduced forests is endowed with a combinatorial Hopf algebra structure. The corresponding structure on the space of planar rooted hyperforests is also described.
There is a one-to-one correspondence between natural numbers and rooted trees; the number is called the Matula number of the rooted tree. We show how a large number of properties of trees can be obtained directly from the corresponding Matula number.
For a labeled, rooted tree with edges oriented towards the root, we consider the vertices as parking spots and the edge orientation as a one-way street. Each driver, starting with her preferred parking spot, searches for and parks in the first unoccupied spot along the directed path to the root. If all $n$ drivers park, the sequence of spot preferences is called a parking function. We consider the sequences, called \emph{prime} parking functions, for which each driver parks and each edge in the tree is traversed by some driver after failing to park at her preferred spot. We prove that the tota…
We show that the space of rooted tree-based phylogenetic networks is connected under rooted nearest-neighbour interchange (rNNI) moves.
In this paper we give a sufficient and necessary condition for two rooted trees with the same plucking polynomial. Furthermore, we give a criteria for a sequence of non-negative integers to be realized as a rooted tree.
Let T be the category whose objects are rooted trees and morphisms are order embeddings preserving the root. We prove that finitely generated representations of T are Noetherian using techniques developed by Sam and Snowden which generalize classical Grobner theory. The proof uses a relative version of Kruskals tree Theorem.
Richard P. Stanley conjectured that finite trees can be distinguished by their chromatic symmetric functions. In this paper, we prove an analogous statement for posets: Finite rooted trees can be distinguished by their order quasisymmetric functions.
Rooted tree maps assign to an element of the Connes-Kreimer Hopf algebra of rooted trees a linear map on the noncommutative polynomial algebra in two letters. Evaluated at any admissible word these maps induce linear relations between multiple zeta values. In this note we show that the derivation relations for multiple zeta values are contained in this class of linear relations.
Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the second named author introduced rooted tree maps as a family of linear maps on the noncommutative polynomial algebra in two letters. These give a class of relations among multiple zeta values, which are known to be a subclass of the so-called linear part of the Kawashima relations. In this paper we show the opposite implication, that is the linear part of the Kawashima relations is implied by the relations coming from rooted tree maps.
We consider the number of nodes in the levels of unlabelled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned Galton-Watson trees and forests to the case of unlabelled rooted trees and show that they behave in this respect essentially like a conditioned Galton-Watson process.
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