scholarly journals The Vertical Profile of Embedded Trees

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.

scholarly journals A Construction of a β-Coalescent via the Pruning of Binary Trees

2013 ◽  
Vol 50 (3) ◽  
pp. 772-790 ◽  
Author(s):  
Romain Abraham ◽  
Jean-François Delmas
Keyword(s):  
State Space ◽  
Binary Tree ◽  
Time Change ◽  
Random Tree ◽  
Binary Trees ◽  
Continuum Random Tree ◽  
Coalescent Process ◽  
Continuous State Space ◽  
Continuous State ◽  
Continuous Analogue

Considering a random binary tree with n labelled leaves, we use a pruning procedure on this tree in order to construct a β(3/2,1/2)-coalescent process. We also use the continuous analogue of this construction, i.e. a pruning procedure on Aldous's continuum random tree, to construct a continuous state space process that has the same structure as the β-coalescent process up to some time change. These two constructions enable us to obtain results on the coalescent process, such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event.

scholarly journals A Construction of a β-Coalescent via the Pruning of Binary Trees

2013 ◽  
Vol 50 (03) ◽  
pp. 772-790 ◽  
Author(s):  
Romain Abraham ◽  
Jean-François Delmas
Keyword(s):  
State Space ◽  
Binary Tree ◽  
Time Change ◽  
Random Tree ◽  
Binary Trees ◽  
Continuum Random Tree ◽  
Coalescent Process ◽  
Continuous State Space ◽  
Continuous State ◽  
Continuous Analogue

Considering a random binary tree withnlabelled leaves, we use a pruning procedure on this tree in order to construct a β(3/2,1/2)-coalescent process. We also use the continuous analogue of this construction, i.e. a pruning procedure on Aldous's continuum random tree, to construct a continuous state space process that has the same structure as the β-coalescent process up to some time change. These two constructions enable us to obtain results on the coalescent process, such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event.

scholarly journals perfectphyloR: An R package for reconstructing perfect phylogenies

2019 ◽  
Vol 20 (1) ◽  
Author(s):  
Charith B. 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

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.

scholarly journals Reducibility and Solvability of Some Classes of Kryuchkov Binary Tree Pairs

10.37236/2028 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
Maria Madonia ◽  
Giuseppe Scollo
Keyword(s):  
Induction Hypothesis ◽  
Binary Tree ◽  
Binary Trees ◽  
Equal Size ◽  
Future Research ◽  
Theoretic Approach ◽  
Research Directions ◽  
Open Questions ◽  
Future Research Directions

This paper addresses the problem of characterizing classes of pairs of binary trees of equal size for which a signed reassociation sequence, in the Eliahou-Kryuchkov sense, can be shown to exist, either with a size induction hypothesis (reducible pairs), or without it (solvable pairs). A few concepts proposed by Cooper, Rowland and Zeilberger, in the context of a language-theoretic approach to the problem, are here reformulated in terms of signed reassociation sequences, and some of their results are recasted and proven in this framework. A few strategies, tactics and combinations thereof for signed reassociation are introduced, which prove useful to extend the results obtained by the aforementioned authors to new classes of binary tree pairs. In particular, with reference to path trees, i.e. binary trees that have a leaf at every level, we show the reducibility of pairs where (at least) one of the two path trees has a triplication at the first turn below the top level, and we characterize a class of weakly mutually crooked path tree pairs that are neither reducible nor solvable by any previously known result, but prove solvable by appropriate reassociation strategies. This class also includes a subclass of mutually crooked path tree pairs. A summary evaluation of the achieved results, followed by an outline of open questions and future research directions conclude the paper.

Binary Trees and the n-Cutset Property

1991 ◽  
Vol 34 (1) ◽  
pp. 23-30 ◽  
Author(s):  
Peter Arpin ◽  
John Ginsburg
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Binary Tree ◽  
Ordered Set ◽  
Maximal Chain ◽  
Binary Trees ◽  
Complete Binary Tree ◽  
Maximal Elements ◽  
Extremal Case ◽  
Result Theorem

AbstractA partially ordered set P is said to have the n-cutset property if for every element x of P, there is a subset S of P all of whose elements are noncomparable to x, with |S| ≤ n, and such that every maximal chain in P meets {x} ∪ S. It is known that if P has the n-cutset property then P has at most 2n maximal elements. Here we are concerned with the extremal case. We let Max P denote the set of maximal elements of P. We establish the following result. THEOREM: Let n be a positive integer. Suppose P has the n-cutset property and that |Max P| = 2n. Then P contains a complete binary tree T of height n with Max T = Max P and such that C ∩ T is a maximal chain in T for every maximal chain C of P. Two examples are given to show that this result does not extend to the case when n is infinite. However the following is shown. THEOREM: Suppose that P has the ω-cutset property and that |Max P| = 2ω. If P — Max P is countable then P contains a complete binary tree of height ω

Some Monotone Almost Balanced Knockout Tournament Plans

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.

scholarly journals Connections on Valuated Binary Tree and Their Applications in Factoring Odd Integers

2021 ◽  
pp. 134-153
Author(s):  
Xingbo Wang ◽  
Jinfeng Luo ◽  
Ying Tian ◽  
Li Ma
Keyword(s):  
Binary Tree ◽  
Numerical Experiments ◽  
Mathematical Reasoning ◽  
Binary Trees ◽  
Integer Factorization ◽  
Parallel Connection ◽  
Central Lines ◽  
Factorization Problem ◽  
Integer Factorization Problem

This paper makes an investigation on geometric relationships among nodes of the valuated binary trees, including parallelism, connection and penetration. By defining central lines and distance from a node to a line, some intrinsic connections are discovered to connect nodes between different subtrees. It is proved that a node out of a subtree can penetrate into the subtree along a parallel connection. If the connection starts downward from a node that is a multiple of the subtree’s root, then all the nodes on the connection are multiples of the root. Accordingly composite odd integers on such connections can be easily factorized. The paper proves the new results with detail mathematical reasoning and demonstrates several numerical experiments made with Maple software to factorize rapidly a kind of big odd integers that are of the length from 59 to 99 decimal digits. It is once again shown that the valuated binary tree might be a key to unlock the lock of the integer factorization problem.

scholarly journals A Combinatorial Bijection on di-sk Trees

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.

scholarly journals Parallelizing DSW Trees

2019 ◽  
Author(s):  
Willard Stanley
Keyword(s):  
Binary Tree ◽  
Binary Trees ◽  
Processing Elements ◽  
Do So

A parallelization of the Day-Stout-Warren algorithm for balancing binary trees. As its input, this algorithm takes an arbitrary binary tree and returns an equivalent tree which is balanced so as to preserve the θ(log(n)) lookup time for elements of the tree. The sequential Day-Stout-Warren algorithm has a linear runtime and uses constant space. This new parallelization of the Day-Stout-Warren algorithm attempts to do the same while providing a speedup which is as near as possible to linear to the number of processing elements. Also, ideally it should do so in an online fashion, without blocking new reads, inserts, and deletes.

scholarly journals Design of Adaptive Compression Algorithm Elias Delta Code and Huffman

2018 ◽  
Author(s):  
Andysah Putera Utama Siahaan
Keyword(s):  
Data Compression ◽  
Binary Tree ◽  
Research Data ◽  
Input String ◽  
Binary Trees ◽  
Text File ◽  
Compression Process ◽  
Storage Media ◽  
Huffman Algorithm ◽  
Adaptive Compression

Compression aims to reduce data before storing or moving it into storage media. Huffman and Elias Delta Code are two algorithms used for the compression process in this research. Data compression with both algorithms is used to compress text files. These two algorithms have the same way of working. It starts by sorting characters based on their frequency, binary tree formation and ends with code formation. In the Huffman algorithm, binary trees are formed from leaves to roots and are called tree-forming from the bottom up. In contrast, the Elias Delta Code method has a different technique. Text file compression is done by reading the input string in a text file and encoding the string using both algorithms. The compression results state that the Huffman algorithm is better overall than Elias Delta Code.

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