Supersolvable LL-lattices of binary trees
Riccardo Biagioli, Frederic Chapoton
证明了某些二叉树偏序集是超可解格，并给出了显式EL-标号。计算它们的特征多项式，以不同的方式恢复它们的已知因式分解。
Some posets of binary leaf-labeled trees are shown to be supersolvable lattices and explicit EL-labelings are given. Their characteristic polynomials are computed, recovering their known factorization in a different way.
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How is a Chordal Graph like a Supersolvable Binary Matroid?
设G是有限单图。从R. P. Stanley已知G的循环拟阵是超可解的当且仅当G是弦（刚性）的：这是阅读弦图上的狄拉克定理的另一种方式。弦二元拟阵一般不是超可解的。然而，我们证明了，对于每一个超可解二元拟阵M，M的模平坦的极大链规范地确定弦图。
A matroid is supersolvable if it has a maximal chain of flats each of which is modular. A matroid is saturated if every round flat is modular. In this article we present supersolvable saturated matroids as analogues to chordal graphs, and we show that several results for chordal graphs hold in this matroid context. In particular, we consider matroid analogues of the reduced clique graph and clique trees for chordal graphs. The latter is a maximum-weight spanning tree of the former. We also show that the matroid analogue of a clique tree is an optimal decomposition for the matroid parameter of …
A simple bijection between binary trees and colored ternary trees
在这篇短文中，我们首先给出了二叉树和有色三元树之间的一个简单的双射，然后导出了一个与广义Catalan数有关的新恒等式。
In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare these results to analogous work for contiguous tree patterns. Next, we give an explicit generating function that counts binary trees avoiding a single non-contiguous tree pattern according to number of leaves. In addition, we enumerate binary trees that simultaneously avoid more than one tree pattern. Finally, we explore connections between pattern-avoiding t…
Horton-Strahler numbers for binary butterfly trees: exact analysis
本文针对Peca提出的二元蝶形树Horton-Strahler数问题，通过组合分析证明该类由两棵二叉树特殊粘合得到的树，其渐近性质与经典二叉树一致。
On the Hook Length Formula for Binary Trees
我们提出了一个简单的组合证明Postnikov的钩长公式的二叉树。
Let $Ω_n$ be the family of binary trees on $n$ vertices obtained by identifying the root of an rgood binary tree with a vertex of maximum eccentricity of a binary caterpillar. In the paper titled "On different middle parts of a tree (The electronic journal of combinatorics, 25 (2018), no. 3, paper 3.17, 32 pp)", Smith et al. conjectured that among all binary trees on $n$ vertices the pairwise distance between any two of center, centroid and subtree core is maximized by some member of the family $Ω_n$. We first obtain the rooted binary tree which minimizes the number of root containing subtrees…
New hook length formulas for binary trees
我们发现了两个新的二叉树的钩长公式。我们的公式的特殊性在于钩长$h_v$以指数形式出现。
In this paper, we present a new method for determining the optimal pebbling number of a complete binary tree. This method reveals a curious connection between the optimal pebbling numbers of complete binary trees and the Conolly-Fox sequence, a type of meta-Fibonacci sequence.
An analogue of the plactic monoid for binary search trees
我们引入了一个幺半群结构上的一组特定的标记二叉树，通过一个类似的过程的建设的plastic幺半群。这导致一个新的解释的代数平面二叉树的Loday-Ronco。
Hex-trees are identified as a particular instance of weighted unary-binary trees. The Horton-Strahler numbers of these objects are revisited, and, thanks to a substitution that is not immediately intuitive, explicit results are possible. They are augmented by asymptotic evaluations as well. Furthermore, marked ordered trees (in bijection to skew Dyck paths) are investigated, followed 3-Motzkin paths and multi-edge trees. The underlying theme is sequence A002212 in the Encyclopedia of integer sequences.
We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T, T'] where T and T' are balanced binary trees are isomorphic as posets to a hypercube. We introduce synchronous grammars that allow to generate tree-like structures and obtain fixed-point functional equations to enumerate these. We also introduce imbalance tree patterns and show that they can be used to describe some sets of balanced binary trees that play a particular role in the Tamari lattice. Finally, we investigate other families of binary trees that are also close…
On the rotation distance between binary trees
我们开发了用于计算二叉树之间的旋转距离的组合方法，即，等效地，多边形的三角剖分之间的翻转距离。作为一个应用，我们证明了对于每个n，在距离为2n-O（sqrt（n））处存在大小为n的树.
A Hopf operad of forests of binary trees and related finite-dimensional algebras
这篇文章的主题是叶标根二叉树的代数组合学和这种树的森林。一个霍普夫操作数的结构被定义在由叶子标记的、有根的、二叉树的森林所构成的向量空间上。利用森林上的偏序集，给出了上积及其对偶积的一个显式公式。
We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T0, T1] where T0 and T1 are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and synchronous grammars to get a functional equation of the generating series enumerating balanced tree intervals.
Yet another generalization of Postnikov's hook length formula for binary trees
我们发现另一个单参数推广Postnikov的钩长公式的二叉树。我们的公式的特殊性在于钩长$h_v$以指数形式出现。作为一个应用，我们得到了另一个简单的二叉树的钩长公式时，底层参数的值为1/2。
In analogy to other concepts of a similar nature, we define the inducibility of a rooted binary tree. Given a fixed rooted binary tree $B$ with $k$ leaves, we let $γ(B,T)$ be the proportion of all subsets of $k$ leaves in $T$ that induce a tree isomorphic to $B$. The inducibility of $B$ is $\limsup_{|T| \to \infty} γ(B,T)$. We determine the inducibility in some special cases, show that every binary tree has positive inducibility and prove that caterpillars are the only binary trees with inducibility $1$. We also formulate some open problems and conjectures on the inducibility. Finally, we pres…
This paper considers the enumeration of trees avoiding a contiguous pattern. We provide an algorithm for computing the generating function that counts n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this counting mechanism, we study the analogue of Wilf equivalence in which two tree patterns are equivalent if the respective n-leaf trees that avoid them are equinumerous. We investigate the equivalence classes combinatorially. Toward establishing bijective proofs of tree pattern equivalence, we develop a general method of restructuring trees that conjecturally succeeds …
It is proved that the average number of segments on the right branch of a binary tree of size n tends to 3 as n tends to $\infty$. Also the fraction of trees with k segments on the right branch from all trees of size n tends to $\frac{k}{2^{k+1}}$ as n tends to $\infty$
A classical problem in phylogenetic tree analysis is to decide whether there is a phylogenetic tree $T$ that contains all information of a given collection $\cP$ of phylogenetic trees. If the answer is "yes" we say that $\cP$ is compatible and $T$ displays $\cP$. This decision problem is NP-complete even if all input trees are quartets, that is binary trees with exactly four leaves. In this paper, we prove a sufficient condition for a set of binary phylogenetic trees to be compatible. That result is used to give a short and self-contained proof of the known characterization of quartet sets of …
We take a global view at substitution invariant Sturmian sequences. We show that homogeneous substitution invariant Sturmian sequences $s_{α,α}$ can be indexed by two binary trees, associated directly to Johannes Kepler's tree of harmonic fractions from 1619. We obtain similar results for the inhomogeneous sequences $s_{α,1-α}$ and $s_{α,0}$.
We show that for every $k$, the probability that a randomly selected vertex of a random binary search tree on $n$ nodes is at distance $k-1$ from the closest leaf converges to a rational constant $c_k$ as $n$ goes to infinity.
An earlier characterization of topologically ordered (lexicographic) path-length sequences of binary trees is reformulated in terms of an integrality condition on a scaled Kraft sum of certain subsequences (full segments, or islands). The scaled Kraft sum is seen to count the set of ancestors at a certain level of a set of topologically consecutive leaves is a binary tree.
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。
The Algebra of Binary Search Trees
我们引入一个幺半群结构的二叉搜索树的集合上，由一个过程非常类似的建设的plastic幺半群，罗宾逊-Schensted插入被替换的二叉搜索树插入。这导致了一个新的结构的代数的平面二叉树的洛迪-朗科，定义它以同样的方式作为非交换对称函数和自由对称函数。我们简要地解释如何的主要已知性质的Loday-Ronco代数可以描述和证明与此组合的观点，然后讨论它从表示理论的角度来看，这反过来又导致新的组合性质的二叉树。
Two involutions on binary trees and generalizations
本文研究了二叉树上的两个对合及其推广，结合经典双射$φ$解决了多个公开的双射问题，并导出了有根树上的新统计量与四元等分布。
The existence of greatest lower bounds in the imbalance order of path-length sequences of binary trees is seen to be a consequence of a joint monotonicity property of the greater and lower expension operations. Path length sequences that are join-irreducible in the imbalance lattice are characterized.
Networks bijective to permutations
本文研究由源点、汇点和中性点组成的与排列双射的网络，通过多米诺或Rothe图构造边集，引入森林对象证明其与逆排列网络双射，并证明网络偏序集为有限分次格且具有$EL$-标号，进而证明超可解性并计算Möbius函数。
We study the set of NBC sets (no broken circuit sets) of the Linial arrangement and deduce a constructive bijection to the set of local binary search trees. We then generalize this construction to two families of Linial type arrangements for which the bijections are with some $k$-ary labelled trees that we introduce for this purpose.
Bijections in weakly increasing trees via binary trees
该研究受平面树对称性启发，通过交换对应二叉树指定节点的左右孩子角色，构建了四个弱递增树双射，为相关对称性提供双射证明并给出Deutsch平面树双射的非递归构造，还探讨了其在排列模式和统计中的应用。
Bijective Proofs of Identities from Colored Binary Trees
本文给出了Sun最近提出的两个涉及广义Catalan数$C_{m,5}(n)={m\over 5n+m}{5n+m\choose n}$的组合恒等式的双射证明.
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.
A q-analogue of the path length of binary search trees
重新制定的二叉搜索树的路径长度给出的排列，允许扩展的定义的实例的话，其中的字母是由独立的几何随机变量（参数q）。用这种方法得到了期望和方差的表达式，在极限情况下，这些表达式对于$q\to1$是经典表达式。
One wishes to remove $k-1$ edges of a vertex-weighted tree $T$ such that the weights of the $k$ induced connected components are approximately the same. How well can one do it ? In this paper, we investigate such $k$-separator for {\em quasi-binary} trees. We show that, under certain conditions on the total weight of the tree, a particular $k$-separator can be constructed such that the smallest (respectively the largest) weighted component is lower (respectively upper) bounded. Examples showing optimality for the lower bound are also given.
The register function (or Horton-Strahler number) of a binary tree is a well-known combinatorial parameter. We study a reduction procedure for binary trees which offers a new interpretation for the register function as the maximal number of reductions that can be applied to a given tree. In particular, the precise asymptotic behavior of the number of certain substructures ("branches") that occur when reducing a tree repeatedly is determined. In the same manner we introduce a reduction for simple two-dimensional lattice paths from which a complexity measure similar to the register function can …
Labeled Chip-firing on Binary Trees with $2^n-1$ Chips
研究二叉树上带 $2^n-1$ 个芯片的标记芯片发射，证明终止构型的排序性质及终局移动偏序集为模格。
Meta-Fibonacci Sequences, Binary Trees, and Extremal Compact Codes
我们看看一个家庭的元斐波那契序列中出现的研究叶数在最大的水平在某些无穷序列的二叉树，限制组成的整数，和二进制紧凑代码。对于这个家庭的元斐波那契序列和两个家庭的相关序列，我们得到普通的生成函数和递归关系。在这些序列家族中包括在线序列百科全书（OEIS）中的几个众所周知的序列。
Binary Search Tree insertion, the Hypoplactic insertion, and Dual Graded Graphs
Fomin（1994）在同一顶点集上的两个分次图之间引入了对偶的概念。他还介绍了一个推广到双分级图的经典罗宾逊-申斯特-克努特算法。我们将展示如何Fomin的方法适用于二叉搜索树插入算法，也被称为西尔维斯特插入，和hypoplastic插入算法。
The register function (or Horton-Strahler number) of a binary tree is a well-known combinatorial parameter. We study a reduction procedure for binary trees which offers a new interpretation for the register function as the maximal number of reductions that can be applied to a given tree. In particular, the precise asymptotic behavior of the number of certain substructures ("branches") that occur when reducing a tree repeatedly is determined. In the same manner we introduce a reduction for simple two-dimensional lattice paths from which a complexity measure similar to the register function can …
We here both unify and generalize nonassociative structures on typed binary trees, that is to say plane binary trees which edges are decorated by elements of a set $Ω$. We prove that we obtain such a structure, called an $Ω$-dendriform structure, if $Ω$ has four products satisfying certain axioms (EDS axioms), including the axioms of a diassociative semigroup. This includes matching dendriform algebras introduced by Zhang, Gao and Guo and family dendriform algebras associated to a semigroup introduced by Zhang, Gao and Manchon , and of course dendriform algebras when $Ω$ is reduced to a single…
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。
There is a unique path from the root of a tree to any other vertex. Every vertex, except the root, has a parent: the adjoining vertex on this unique path. This is the conventional definition of the parent vertex. For complete binary trees, however, we show that it is useful to define another parent vertex, called a \emph{distant parent}. The study of distant parents leads to novel connections with dyadic rational numbers. Moreover, we apply the concepts of close and distant parent vertices to deduce an apparently new sense in which continued fractions are `best' rational approximations.
A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of complete binary trees whose leaves are labeled by letters of an alphabet containing at least three letters a function is congruence preserving if and only if it is polynomial. This exhibits an example of a non commutative and non associative affine complete algebra. As far as we know, it is the first example of such an algebra.
Alternating Extremes in Graceful Labelings of Full Binary Trees and Spider Trees
研究完全二叉树和蜘蛛树优美标号中固定形式的交替极值问题，证明了梳状完全二叉树的固定脊柱猜想并计算验证相关完全二叉树，证明蜘蛛树自匹配腿填充定理，给出相关计算结果并表述六臂问题。
We give a new construction of a Hopf algebra defined first by Reading whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e., Baxter permutations, pairs of twin binary trees, etc.). Our construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like correspondence and insertion algorithm. Indeed, the Baxter monoid leads to the definition of a lattice structure over pairs of twin binary trees and the definition of a Hopf algebra. The algebraic properties of this Hopf algebra are st…
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…
Fringe trees of Patricia tries, compressed binary search trees, and three other random full binary trees
研究Patricia tries和压缩二叉搜索树中边树的分布及其渐近性质，并与其他随机满二叉树模型进行比较。
We present a new uniform random sampler for binary trees with $n$ internal nodes consuming $2n + Θ(\log(n)^2)$ random bits on average. This makes it quasi-optimal and out-performs the classical Remy algorithm. We also present a sampler for unary-binary trees with $n$ nodes taking $Θ(n)$ random bits on average. Both are the first linear-time algorithms to be optimal up to a constant.
Upper Bounds on the Average Height of Random Binary Trees
本研究推广了Devroye的结果，证明了叶中心二叉树源生成的随机树的平均高度上界为$\mathcal{O}(\log n)$。
A tree-based network $N$ on $X$ is called universal if every phylogenetic tree on $X$ is a base tree for $N$. Recently, binary universal tree-based networks have attracted great attention in the literature and their existence has been analyzed in various studies. In this note, we extend the analysis to non-binary networks and show that there exist both a rooted and an unrooted non-binary universal tree-based network with $n$ leaves for all positive integers $n$.
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.
A binary tree representation for the 2-adic valuation of a sequence arising from a rational integral
我们提出了一个二叉树，描述了2-adic估值的一个序列的系数所产生的评价的合理的积分。
In this paper we examine planted binary plane trees. First, we provide an exact formula for the number of planted binary trees with given Horton-Strahler orders. Then, using the notion of entropy, we examine the structural complexity of random planted binary trees with N vertices. Finally, we quantify the complexity of the tree's structural properties as tree grows in size, by evaluating the entropy rate for planted binary plane trees with N vertices and for planted binary plane trees that satisfy Horton Law with Horton exponent R.
Order structure on the algebra of permutations and of planar binary trees
在“平面二叉树的霍普夫代数”，Adv.Math.139（1998），no.2，293- 309中，我们通过归纳法构造了由平面二叉树生成的向量空间上的分次结合积（分别为：排列）。在本文中，我们证明了一个明确的公式，这种产品的偏序的集合上的平面二叉树（分别。对称群上的弱Bruhat序）。
Planar binary trees, noncrossing partitions and the operator-valued S-transform
本文通过平面二叉树与非交叉划分的双射，在算子值框架下重新证明了Voiculescu S-变换的扭曲乘性性质。
This paper describes the shortest path problem in weighted graphs and examines the differences in efficiency that occur when using Dijkstra's algorithm with a Fibonacci heap, binary heap, and self-balancing binary tree. Using C++ implementations of these algorithm variants, we find that the fastest method is not always the one that has the lowest asymptotic complexity. Reasons for this are discussed and backed with empirical evidence.
A fringe subtree of a rooted tree is a subtree consisting of one of the nodes and all its descendants. In this paper, we are specifically interested in the number of non-isomorphic trees that appear in the collection of all fringe subtrees of a binary tree. This number is analysed under two different random models: uniformly random binary trees and random binary search trees. In the case of uniformly random binary trees, we show that the number of non-isomorphic fringe subtrees lies between $c_1n/\sqrt{\ln n}(1+o(1))$ and $c_2n/\sqrt{\ln n}(1+o(1))$ for two constants $c_1 \approx 1.0591261434$…
Three standard subtree transfer operations for binary trees, used in particular for phylogenetic trees, are: tree bisection and reconnection ($TBR$), subtree prune and regraft ($SPR$) and rooted subtree prune and regraft ($rSPR$). For a pair of leaf-labelled binary trees with $n$ leaves, the maximum number of such moves required to transform one into the other is $n-Θ(\sqrt{n})$, extending a result of Ding, Grunewald and Humphries. We show that if the pair is chosen uniformly at random, then the expected number of moves required to transfer one into the other is $n-Θ(n^{2/3})$. These results m…
Phylogenetic networks are a generalization of phylogenetic trees allowing for the representation of non-treelike evolutionary events such as hybridization. Typically, such networks have been analyzed based on their `level', i.e. based on the complexity of their 2-edge-connected components. However, recently the question of how `treelike' a phylogenetic network is has become the center of attention in various studies. This led to the introduction of \emph{tree-based networks}, i.e. networks that can be constructed from a phylogenetic tree, called the \emph{base tree}, by adding additional edges…
Combinatorial generation via permutation languages. VI. Binary trees
本文提出二叉树广义模式规避概念，利用排列编码和 Hartung-Hoang-Mütze-Williams 框架生成各类规避模式二叉树，并建立其与网格排列模式、格路及集合划分的对应关系。
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