On the Automorphism Group of the One-Rooted Binary Tree

Abstract Background A perfect phylogeny is a rooted binary tree that recursively partitions sequences. The nested partitions of a perfect phylogeny provide insight into the pattern of ancestry of genetic sequence data. For example, sequences may cluster together in a partition indicating that they arise from a common ancestral haplotype. Results We present an R package to reconstruct the local perfect phylogenies underlying a sample of binary sequences. The package enables users to associate the reconstructed partitions with a user-defined partition. We describe and demonstrate the major functionality of the package. Conclusion The package should be of use to researchers seeking insight into the ancestral structure of their sequence data. The reconstructed partitions have many applications, including the mapping of trait-influencing variants.

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Some Monotone Almost Balanced Knockout Tournament Plans

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800002977 ◽

1993 ◽

Vol 7 (3) ◽

pp. 361-368

Author(s):

R. W. Chen ◽

F. K. Hwang ◽

Y. C. Yao ◽

A. Zame

Keyword(s):

Real World ◽

General Class ◽

Binary Tree ◽

The Real ◽

Random Labeling ◽

Knockout Tournament ◽

Rooted Binary Tree

Knockout tournaments are often used in sports (or experiments where preferences are registered by comparisons instead of measurements) to determine the champion of an event. A knockout tournament plan (KTP) for n players is a rooted binary tree with n leaves to be labeled by the n players. Each subtree of two leaves represents a match between the two players labeling the two leaves; the winner of the match then moves on to label the root of the subtree. While there are many KTPs to choose from for a given number of players, in the real world an almost balanced KTP is usually chosen. One reason could be the perception that a balanced KTP is “fair” to the players, in the sense that, given a random labeling of leaves by players, a stronger player has a better chance to win the tournament. Surprisingly, it has been shown that not all KTPs have this property, and it is difficult to prove this property for any general class of KTPs. So far the property has been shown to hold only for balanced KTPs. In this paper we extend it to some classes of almost balanced KTPs.

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Automorphism group of the one-divisor graph over the semigroup of upper triangular matrices

Journal of Algebra and Its Applications ◽

10.1142/s0219498821502145 ◽

2020 ◽

pp. 2150214

Author(s):

Xinlei Wang ◽

Dein Wong ◽

Fenglei Tian

Keyword(s):

Automorphism Group ◽

Zero Divisor ◽

Symmetric Groups ◽

Directed Edge ◽

Triangular Matrices ◽

Upper Triangular Matrices ◽

Zero Divisor Graph ◽

Vertex Set ◽

The One ◽

Definition Of

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] a positive integer, [Formula: see text] the semigroup of all [Formula: see text] upper triangular matrices over [Formula: see text] under matrix multiplication, [Formula: see text] the group of all invertible matrices in [Formula: see text], [Formula: see text] the quotient group of [Formula: see text] by its center. The one-divisor graph of [Formula: see text], written as [Formula: see text], is defined to be a directed graph with [Formula: see text] as vertex set, and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text] and [Formula: see text] are, respectively, a left divisor and a right divisor of a rank one matrix in [Formula: see text]. The definition of [Formula: see text] is motivated by the definition of zero-divisor graph [Formula: see text] of [Formula: see text], which has vertex set of all nonzero zero-divisors in [Formula: see text] and there is a directed edge from a vertex [Formula: see text] to a vertex [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text]. The automorphism group of zero-divisor graph [Formula: see text] of [Formula: see text] was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl. 465 (2015) 214–220.]. In this paper, we characterize the automorphism group of one-divisor graph [Formula: see text] of [Formula: see text], proving that [Formula: see text], where [Formula: see text] is the automorphism group of field [Formula: see text], [Formula: see text] is a direct product of some symmetric groups. Besides, an application of automorphisms of [Formula: see text] is given in this paper.

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On automorphism groups of low complexity subshifts

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.70 ◽

2015 ◽

Vol 36 (1) ◽

pp. 64-95 ◽

Cited By ~ 14

Author(s):

SEBASTIÁN DONOSO ◽

FABIEN DURAND ◽

ALEJANDRO MAASS ◽

SAMUEL PETITE

Keyword(s):

Finite Group ◽

Automorphism Group ◽

Polynomial Complexity ◽

Automorphism Groups ◽

Low Complexity ◽

Topological Dynamics ◽

Main Technique ◽

Shift Map ◽

The One ◽

A Finite Group

In this article, we study the automorphism group$\text{Aut}(X,{\it\sigma})$of subshifts$(X,{\it\sigma})$of low word complexity. In particular, we prove that$\text{Aut}(X,{\it\sigma})$is virtually$\mathbb{Z}$for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a$d$-step nilsystem is nilpotent of order$d$and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually$\mathbb{Z}$.

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The Vertical Profile of Embedded Trees

The Electronic Journal of Combinatorics ◽

10.37236/2150 ◽

2012 ◽

Vol 19 (3) ◽

Author(s):

Mireille Bousquet-Mélou ◽

Guillaume Chapuy

Keyword(s):

Binary Tree ◽

Vertical Profile ◽

Random Tree ◽

Binary Trees ◽

Number Of Children ◽

Rooted Binary Tree

Consider a rooted binary tree with $n$ nodes. Assign with the root the abscissa 0, and with the left (resp. right) child of a node of abscissa $i$ the abscissa $i-1$ (resp. $i+1$). We prove that the number of binary trees of size $n$ having exactly $n_i$ nodes at abscissa $i$, for $l \leq i \leq r$ (with $n = \sum_i n_i$), is $$ \frac{n_0}{n_l n_r} {{n_{-1}+n_1} \choose {n_0-1}} \prod_{l\le i\le r \atop i\not = 0}{{n_{i-1}+n_{i+1}-1} \choose {n_i-1}}, $$ with $n_{l-1}=n_{r+1}=0$. The sequence $(n_l, \dots, n_{-1};n_0, \dots n_r)$ is called the vertical profile of the tree. The vertical profile of a uniform random tree of size $n$ is known to converge, in a certain sense and after normalization, to a random mesure called the integrated superbrownian excursion, which motivates our interest in the profile. We prove similar looking formulas for other families of trees whose nodes are embedded in $Z$. We also refine these formulas by taking into account the number of nodes at abscissa j whose parent lies at abscissa $i$, and/or the number of vertices at abscissa i having a prescribed number of children at abscissa $j$, for all $i$ and $j$. Our proofs are bijective.

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A Combinatorial Bijection on di-sk Trees

The Electronic Journal of Combinatorics ◽

10.37236/10484 ◽

2021 ◽

Vol 28 (4) ◽

Author(s):

Shishuo Fu ◽

Zhicong Lin ◽

Yaling Wang

Keyword(s):

Binary Tree ◽

Number Of Components ◽

Symmetric Extension ◽

Alternative Approach ◽

Tree Traversal ◽

Separable Permutations ◽

Rooted Binary Tree ◽

Schur Positive

A di-sk tree is a rooted binary tree whose nodes are labeled by $\oplus$ or $\ominus$, and no node has the same label as its right child. The di-sk trees are in natural bijection with separable permutations. We construct a combinatorial bijection on di-sk trees proving the two quintuples $(\mathrm{LMAX},\mathrm{LMIN},\mathrm{DESB},\mathsf{iar},\mathsf{comp})$ and $(\mathrm{LMAX},\mathrm{LMIN},\mathrm{DESB},\mathsf{comp},\mathsf{iar})$ have the same distribution over separable permutations. Here for a permutation $\pi$, $\mathrm{LMAX}(\pi)/\mathrm{LMIN}(\pi)$ is the set of values of the left-to-right maxima/minima of $\pi$ and $\mathrm{DESB}(\pi)$ is the set of descent bottoms of $\pi$, while $\mathsf{comp}(\pi)$ and $\mathsf{iar}(\pi)$ are respectively the number of components of $\pi$ and the length of initial ascending run of $\pi$. Interestingly, our bijection specializes to a bijection on $312$-avoiding permutations, which provides (up to the classical Knuth–Richards bijection) an alternative approach to a result of Rubey (2016) that asserts the two triples $(\mathrm{LMAX},\mathsf{iar},\mathsf{comp})$ and $(\mathrm{LMAX},\mathsf{comp},\mathsf{iar})$ are equidistributed on $321$-avoiding permutations. Rubey's result is a symmetric extension of an equidistribution due to Adin–Bagno–Roichman, which implies the class of $321$-avoiding permutations with a prescribed number of components is Schur positive. Some equidistribution results for various statistics concerning tree traversal are presented in the end.

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PERFORMANCE APPRAISAL OF TREAP AND HEAP SORT ALGORITHMS

Journal of Natural Sciences Engineering and Technology ◽

10.51406/jnset.v18i1.2027 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1-10

Author(s):

A. D. GBADEBO ◽

A. T. AKINWALE ◽

S. AKINLEYE

Keyword(s):

Performance Appraisal ◽

Binary Tree ◽

Ordered Set ◽

Search Time ◽

Search Tree ◽

Binary Search Tree ◽

Complete Binary Tree ◽

Totally Ordered Set ◽

Rooted Binary Tree ◽

Fast Access

The task of storing items to allow for fast access to an item given its key is an ubiquitous problem in many organizations. Treap as a method uses key and priority for searching in databases. When the keys are drawn from a large totally ordered set, the choice of storing the items is usually some sort of search tree. The simplest form of such tree is a binary search tree. In this tree, a set X of n items is stored at the nodes of a rooted binary tree in which some item y ϵ X is chosen to be stored at the root of the tree. Heap as data structure is an array object that can be viewed as a nearly complete binary tree in which each node of the tree corresponds to an element of the array that stores the value in the node. Both algorithms were subjected to sorting under the same experimental environment and conditions. This was implemented by means of threads which call each of the two methods simultaneously. The server keeps records of individual search time which was the basis of the comparison. It was discovered that treap was faster than heap sort in sorting and searching for elements using systems with homogenous properties.

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perfectphyloR: An R package for reconstructing perfect phylogenies

10.1101/674523 ◽

2019 ◽

Author(s):

Charith Bhagya Karunarathna ◽

Jinko Graham

Keyword(s):

Binary Tree ◽

Sequence Data ◽

R Package ◽

Binary Sequences ◽

Ancestral Haplotype ◽

Perfect Phylogeny ◽

Nested Partitions ◽

Genetic Sequence ◽

Insight Into ◽

Rooted Binary Tree

AbstractBackgroundA perfect phylogeny is a rooted binary tree that recursively partitions sequences. The nested partitions of a perfect phylogeny provide insight into the pattern of ancestry of genetic sequence data. For example, sequences may cluster together in a partition indicating that they arise from a common ancestral haplotype.ResultsWe present an R packageperfectphyloRto reconstruct the local perfect phylogenies underlying a sample of binary sequences. The package enables users to associate the reconstructed partitions with a user-defined partition. We describe and demonstrate the major functionality of the package.ConclusionTheperfectphyloRpackage should be of use to researchers seeking insight into the ancestral structure of their sequence data. The reconstructed partitions have many applications, including the mapping of trait-influencing variants.

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