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Catalan Sequence
The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after Eugène Catalan, though they were previously discovered in the 1730s by Minggatu. The -th Catalan number can be expressed directly in terms of the central binomial coefficients by :C_n = \frac = \frac \qquad\textn\ge 0. The first Catalan numbers for are : . Properties An alternative expression for is :C_n = - for n\ge 0\,, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. Another alternative expression is :C_n = \frac \,, which can be directly interpreted in terms of the cycle lemma; see below. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_C_\quad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyck Word
In the theory of formal languages of computer science, mathematics, and linguistics, a Dyck word is a balanced string of brackets. The set of Dyck words forms a Dyck language. The simplest, Dyck-1, uses just two matching brackets, e.g. ( and ). Dyck words and language are named after the mathematician Walther von Dyck. They have applications in the parsing of expressions that must have a correctly nested sequence of brackets, such as arithmetic or algebraic expressions. Formal definition Let \Sigma = \ be the alphabet consisting of the symbols and Let \Sigma^ denote its Kleene closure. The Dyck language is defined as: : \. Context-free grammar It may be helpful to define the Dyck language via a context-free grammar in some situations. The Dyck language is generated by the context-free grammar with a single non-terminal , and the production: : That is, ''S'' is either the empty string () or is " , an element of the Dyck language, the matching ", and an element of the Dyck l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Path
In combinatorics, a lattice path in the -dimensional integer lattice of length with steps in the Set (mathematics), set , is a sequence of Vector (mathematics and physics), vectors such that each consecutive difference v_i - v_ lies in . A lattice path may lie in any Lattice (group), lattice in , but the integer lattice is most commonly used. An example of a lattice path in of length 5 with steps in S = \lbrace (2,0), (1,1), (0,-1) \rbrace is L = \lbrace (-1,-2), (0,-1), (2,-1), (2,-2), (2,-3), (4,-3) \rbrace . North-East lattice paths A North-East (NE) lattice path is a lattice path in \mathbb^2 with steps in S = \lbrace (0,1), (1,0) \rbrace . The (0,1) steps are called North steps and denoted by N s; the (1,0) steps are called East steps and denoted by E s. NE lattice paths most commonly begin at the origin. This convention allows encoding all the information about a NE lattice path L in a single permutation pattern, permutation word. The length of the wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parse Tree
A parse tree or parsing tree (also known as a derivation tree or concrete syntax tree) is an ordered, rooted tree that represents the syntactic structure of a string according to some context-free grammar. The term ''parse tree'' itself is used primarily in computational linguistics; in theoretical syntax, the term ''syntax tree'' is more common. Concrete syntax trees reflect the syntax of the input language, making them distinct from the abstract syntax trees used in computer programming. Unlike Reed-Kellogg sentence diagrams used for teaching grammar, parse trees do not use distinct symbol shapes for different types of constituents. Parse trees are usually constructed based on either the constituency relation of constituency grammars ( phrase structure grammars) or the dependency relation of dependency grammars. Parse trees may be generated for sentences in natural languages (see natural language processing), as well as during processing of computer languages, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,See . or singly connected networkSee . is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tamari Lattice, Trees
Tamari may refer to: * A type of soy sauce Soy sauce (sometimes called soya sauce in British English) is a liquid condiment of China, Chinese origin, traditionally made from a fermentation (food), fermented paste of soybeans, roasted cereal, grain, brine, and ''Aspergillus oryzae'' or ''A ..., produced mainly in the Chūbu region of Japan * Tamari lattice, a mathematical lattice theory named after mathematician Dov Tamari * Tamari Bar, restaurant in Seattle, Washington, U.S. * '' Te tamari no atua'', 1896 oil painting by Paul Gauguin * Tamari, Ibaraki, village in Niihari District, Ibaraki Prefecture, Japan * Tammari language, language spoken in Benin and Togo People People with the surname Tamari * Amiram Tamari (1913–1981) Israeli illustrator and artist * Dov Tamari (1911–2006; born as Bernhard Teitler) German-born Israeli mathematician * Dov Tamari (brigadier general) (born 1936) Israeli military leader * Meir Tamari (1927–2021) Israeli economist * Nehemia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan 4 Leaves Binary Tree Example
Catalan may refer to: Catalonia From, or related to Catalonia: * Catalan language, a Romance language * Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia Places * 13178 Catalan, asteroid #13178, named "Catalan" * Catalán (crater), a lunar crater named for Miguel Ángel Catalán * Çatalan, İvrindi, a village in Balıkesir province, Turkey * Çatalan, Karaisalı, a village in Adana Province, Turkey * Catalan Bay, Gibraltar * Catalan Sea, more commonly known as the Balearic Sea * Catalan Mediterranean System, the Catalan Mountains Facilities and structures * Çatalan Bridge, Adana, Turkey * Çatalan Dam, Adana, Turkey * Catalan Batteries, Gibraltar People * Catalan, Lord of Monaco (1415–1457), Lord of Monaco from 1454 until 1457 * Alfredo Catalán (born 1968), Venezuelan politician * Alex Catalán (born 1968), Spanish filmmaker * Arnaut Catalan (1219–1253), troubador * Diego Catalán (1928–2008), Spanish philo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,See . or singly connected networkSee . is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Tree
In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theory is that a binary tree is a triple , where ''L'' and ''R'' are binary trees or the empty set and ''S'' is a singleton (a single–element set) containing the root. From a graph theory perspective, binary trees as defined here are arborescences. A binary tree may thus be also called a bifurcating arborescence, a term which appears in some early programming books before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of ''binary tree'' to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted. In ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Chain Multiplication
Matrix chain multiplication (or the matrix chain ordering problem) is an optimization problem concerning the most efficient way to multiply a given sequence of matrices. The problem is not actually to ''perform'' the multiplications, but merely to decide the sequence of the matrix multiplications involved. The problem may be solved using dynamic programming. There are many options because matrix multiplication is associative. In other words, no matter how the product is parenthesized, the result obtained will remain the same. For example, for four matrices ''A'', ''B'', ''C'', and ''D'', there are five possible options: :((''AB'')''C'')''D'' = (''A''(''BC''))''D'' = (''AB'')(''CD'') = ''A''((''BC'')''D'') = ''A''(''B''(''CD'')). Although it does not affect the product, the order in which the terms are parenthesized affects the number of simple arithmetic operations needed to compute the product, that is, the computational complexity. The straightforward multiplication of a mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Operator
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation on a set is a binary function that maps every pair of elements of the set to an element of the set. Examples include the familiar arithmetic operations like addition, subtraction, multiplication, set operations like union, complement, intersection. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. A binary function that involves several sets is sometimes also called a ''binary operation''. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Binary operations are the keystone of most structures that are studied in algebra, in particular in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associativity
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replacement for well-formed formula, expressions in Formal proof, logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the Operation (mathematics), operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
