scholarly journals Noncontiguous Pattern Containment in Binary Trees

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Lara Pudwell ◽  
Connor Scholten ◽  
Tyler Schrock ◽  
Alexa Serrato
Keyword(s):  
Functional Equation ◽  
Generating Function ◽  
Binary Tree ◽  
Binary Trees ◽  
Tree Patterns ◽  
Pattern Containment

We consider the enumeration of binary trees containing noncontiguous binary tree patterns. First, we show that any two ℓ-leaf binary trees are contained in the set of all n-leaf trees the same number of times. We give a functional equation for the multivariate generating function for number of n-leaf trees containing a specified number of copies of any path tree, and we analyze tree patterns with at most 4 leaves. The paper concludes with implications for pattern containment in permutations.

scholarly journals Non-Contiguous Pattern Avoidance in Binary Trees

10.37236/2099 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Michael Dairyko ◽  
Lara Pudwell ◽  
Samantha Tyner ◽  
Casey Wynn
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Binary Tree ◽  
Pattern Avoidance ◽  
Binary Trees ◽  
Tree Pattern ◽  
Tree Patterns ◽  
Number Of Leaves ◽  
Pattern Avoiding Permutations

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 and show that there is exactly one Wilf class of k-leaf tree patterns for any positive integer k.  In addition, we give a bijection between between certain sets of pattern-avoiding trees and sets of pattern-avoiding permutations.  Finally, we enumerate binary trees that simultaneously avoid more than one tree pattern.

scholarly journals Balanced binary trees in the Tamari lattice

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Samuele Giraudo
Keyword(s):  
Functional Equation ◽  
Binary Trees ◽  
Tamari Lattice ◽  
Tree Patterns ◽  
Generating Series ◽  
Nous Montrons ◽  
International Audience ◽  
Balanced Tree

International audience We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals $[T_0, T_1]$ where $T_0$ and $T_1$ 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. Nous montrons que l'ensemble des arbres équilibrés est clos par intervalle dans le treillis de Tamari. Nous caractérisons la forme des intervalles du type $[T_0, T_1]$ où $T_0$ et $T_1$ sont équilibrés en montrant qu'en tant qu'ensembles partiellement ordonnés, ils sont isomorphes à un hypercube. Nous introduisons la notion de motif d'arbre et de grammaire synchrone dans le but d'établir une équation fonctionnelle de la série génératrice qui dénombre les intervalles d'arbres équilibrés.

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

A note on the time-dependent and the stationary behaviour of a semi-infinite reservoir subject to a combination of Markovian inflows

1971 ◽  
Vol 8 (04) ◽  
pp. 708-715 ◽  
Author(s):  
Emlyn H. Lloyd
Keyword(s):  
Functional Equation ◽  
Explicit Formula ◽  
Generating Function ◽  
Present Theory ◽  
The Other ◽  
Time Dependent ◽  
Independent Sequence ◽  
Time Dependent Solution ◽  
Infinite Reservoir ◽  
Stationary Behaviour

The present theory of finite reservoirs is not rich in general theorems even when of the simple Moran type, with unit draft and stationary discrete independent-sequence inflows. For the corresponding systems with unbounded capacity however there are two classes of results which have been known for some time. One of these classes is concerned with the time-dependent solution, where the theory provides a functional equation for the generating function of the time to first emptiness (Kendall (1957)), and the other with the asymptotic stationary distribution of reservoir contents, for which an explicit formula for the generating function is available (Moran (1959)).

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (3) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (03) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 &lt; m = h'(1 –) &lt; ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) &lt; 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

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 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 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.

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