Tree diagram

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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.

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Assignmentof Discrete Mathematics
Tree structure
An undirectedgraphis a tree if and only if there is a unique simple path
betweenany two of its vertices.

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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.