Tree 4
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A tree diagram is a connected undirected graph without simple circuits. Trees are used to represent hierarchies and family relationships. A graph is a tree if and only if there is a unique simple path between any two vertices. This is because if there were multiple paths, it would form a circuit, which trees do not have by definition.
Tree 4
- 1. Page 1 Assignmentof Discrete Mathematics Define Tree Diagram A tree is aconnectedundirectedgraphwithno simple circuits. A particular type of graph calleda tree, sonamed because suchgraphs resemble trees. For example, family trees are graphs that represent genealogical charts. Family trees use vertices torepresent the members of afamily and edges to represent parent–childrelationships. The family tree of the male members of the Bernoulli family of Swiss. The undirectedgraphrepresenting afamily tree (restrictedtopeople of just one gender and withno inbreeding) is anexample of a tree.
- 2. Page 2 Assignmentof Discrete Mathematics Tree structure An undirectedgraphis a tree if and only if there is a unique simple path betweenany two of its vertices.
- 3. Page 3 Assignmentof Discrete Mathematics Proof: First assume that T is a tree. Then T is a connectedgraphwithno simple circuits. Let x and y be two verticesof T. Because T is connected, by Theorem1 of Section10.4 there is asimple pathbetweenx and y. Moreover, this pathmust be unique, for if there were asecondsuch path, the path formedby combining the first pathfrom x to y followedby the path from y to x obtainedby reversing the order of the secondpath from x to y would form a circuit. This implies, using that there is a simple circuit inT. Hence, there is a unique simple pathbetween any two vertices of atree. Nowassume that there is a unique simple path betweenany two vertices of agraph T. Then T is connected, because there is a path betweenany two of its vertices. Furthermore, Tcanhave no simple circuits. Tosee that this is true, suppose T had a simple circuit that contained the vertices x andy. Then there wouldbe twosimple paths betweenx and y, because the simple circuit is made up of a simple path from x toy and a second simple pathfrom y to x. Hence, a graph witha unique simple pathbetweenany two verticesis atree. Teams A and B play in a tournament. The team that first wins two games wins the tournament. Find the number of possible ways in which the tournament can occur.