Free step-by-step solutions to Discrete Mathematics with Applications 6th Edition Discrete Mathematics and its Applications, 5th Edition Discrete Mathematics. Discrete mathematics and its applications / Kenneth H. Rosen. — 7th ed. .. Its Applications, published by Pearson, currently in its sixth edition, which has been . Discrete Mathematics and Its Applications (6th edition) – Solutions (1). Uploaded by. Quang Mai. Loading Preview. Sorry, preview is currently unavailable.
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In mathematicsand, more specifically, in graph theory fdition, a tree is an undirected graph in which any two vertices are connected by exactly one path. Every acyclic connected graph is a tree, and vice versa.
A forest is a disjoint union of trees, or equivalently an acyclic graph that is not necessarily connected. 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 disfrete it is called an arborescence branching or out-tree  —or making all its edges point towards the root—in which case it is called an anti-arborescence  or in-tree.
The term “tree” was coined in by the British mathematician Arthur Cayley.
A tree is an undirected graph G that satisfies any of the following equivalent conditions:. If G has finitely many vertices, say n of them, then the above statements are also equivalent to any of the following conditions:.
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As elsewhere in graph theory, the order-zero graph graph with no vertices is generally not considered to be a tree: It may, however, be considered as a forest consisting of zero trees. An internal vertex or inner vertex or branch vertex is a vertex of degree at least 2. Similarly, an external vertex or outer vertexterminal vertex or didcrete is a vertex of degree 1. An irreducible tree or series-reduced tree is a tree in which there is no vertex of degree 2.
A forest is an undirected aoplications, all of whose connected components are trees; in other words, the matg consists of a disjoint union of trees. Equivalently, a forest is an undirected acyclic graph. As special cases, an empty graph, a single tree, and the discrete graph on a set of vertices that is, the graph with these vertices that has no edgesare examples of forests.
A polytree  or oriented tree   or singly connected network  is applciations directed acyclic graph DAG whose underlying undirected diacrete is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic. A directed tree is a directed graph which would be a tree if the directions on the edges were ignored, i. Some authors restrict the phrase to the case where the edges are all directed towards a particular vertex, or all directed away from a particular vertex see arborescence.
A rooted tree is a tree in which one vertex has been designated the root. The edges of a rooted tree can be assigned a natural orientation, either applciations from or towards the discgete, in which case the structure becomes a directed rooted tree.
When a directed rooted tree has an orientation away from the root, it is called an arborescencebranchingor out-tree ; when it has an orientation towards the root, it is called an anti-arborescence or in-tree.
A rooted tree which is a subgraph of some graph G is a normal tree if the ends of every edge in G are comparable in this tree-order whenever those ends are vertices of the tree Diestelp. Rooted trees, often with additional structure such as ordering of the neighbors at each vertex, are a key data structure in computer science; see tree data structure. In a context where trees are supposed to have a root, a tree without any designated root is called a free tree. A labeled tree is a tree in which each vertex is given a unique label.
The vertices of a labeled tree on n vertices are typically given the labels 1, 2, …, n. A recursive tree is a labeled rooted tree where the vertex labels respect the tree order i.
Tree (graph theory)
In a rooted tree, the parent of a vertex is the vertex connected to it on the path to the root; every vertex except the root has a unique parent. A child of a vertex v is a vertex of which v is the parent. A descendant of any vertex v is any vertex which is either the child of v or is recursively the descendant of any of the children of v. A sibling to a vertex v is any other vertex on the tree which has the same parent as v.
The root is an external vertex if it has precisely one child. A leaf is different from the root. The height of a vertex in a rooted tree is the length of the longest downward path to a leaf from that vertex.
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The height of the tree is idscrete height of the root. The depth of a vertex is the length of the path to its root root path. This is commonly needed in the manipulation of the various self-balancing trees, AVL trees in editiln. The root has depth zero, leaves have height zero, and a tree with only a single vertex hence both a root and leaf has depth and height zero.
A k-ary tree is a rooted tree in which each vertex has at most k children. An ordered tree or plane tree is a rooted tree in which an ordering is specified for the children of each vertex. Given an embedding of a applicztions tree in the plane, if one fixes a direction of children, say left to right, then an embedding gives an ordering of the children.
Conversely, given an ordered tree, and conventionally drawing the root ediyion the top, then the child vertices in an ordered tree can be drawn left-to-right, yielding an essentially unique planar embedding. A more idscrete problem is to count spanning trees in an undirected graphwhich is addressed by the matrix applicaations theorem. Cayley’s formula is the special case of spanning trees in a complete graph.
The similar problem of counting all the subtrees regardless of size has been shown to be P-complete in the general case Jerrum Counting the number of unlabeled free trees is a harder problem. No closed formula for the number t n of trees with n vertices up to graph isomorphism is known. The first few values of t n are. This is a consequence of his asymptotic estimate for the number r n of unlabeled rooted applicstions with n vertices:.
Knuthchap. The first few values of r n are . From Wikipedia, the free encyclopedia. Trees A labeled tree with 6 vertices and 5 edges. Combinatorics for Computer Science.
Discrete Mathematics and Its Applications Information Center:
Graph Theoretic Methods in Multiagent Networks. Design and Analysis of Approximation Algorithms. Gross; Jay Yellen; Ping Zhang Handbook of Graph Theory, Second Edition. Theory and Algorithms 5th ed. Algorithms and Data Structures: Sets, Logic and Maths for Computing. Discrete Mathematics and Its Applications, 7th edition. However it should be mentioned that inK. Bauer und Raspe,presented a proof of Euler’s polyhedron theorem which relies on trees on pages 20— He proved the relation via an argument relying mathh trees.
A labeled tree with 6 vertices and 5 edges. Table of graphs and parameters. Wikimedia Commons has media related to Tree graph theory.