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Understanding Trees in Graph Theory: Definitions, Types, and Applications

This article explores the concept of trees in graph theory, defining key terms such as rooted trees, leaves, and internal vertices. It discusses the characteristics of m-ary trees, including binary trees and full m-ary trees. The article delves into the unique path property of trees and provides theorems related to the number of edges and vertices in trees. Applications of trees are highlighted, including genealogy, organizational charts, and computer science. The implications of these properties demonstrate the fundamental role of trees in various fields.

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Understanding Trees in Graph Theory: Definitions, Types, and Applications

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  1. Ch. 10 Trees 10.1 Intro

  2. 4 Examples – trees or not?

  3. Example of ___

  4. Rooted tree

  5. Definitions Def.: • Tree- a connected undirected graph with no simple circuits • Forest- • Rooted tree- tree with a root --i.e. a tree where one vertex is designated as the root and every other edge is directed away from it (ex: descendents, ancestors,…) • Leaf- vertex of tree with no children • Internal vertex-vertex with children

  6. More Def: • m-arytree: a rooted tree where every internal vertex has no more than m children • Special case: m=2 is a binary tree • full m-ary tree: an m-ary tree where every internal vertex has exactly m children.

  7. Full M-arytrees

  8. Q: Is this an M-arytree? (m=__?)Is it a Full M-ary tree?

  9. Is this a full M-ary(if so, m=__?)

  10. Uses of trees • Geneology • Organizational chart • Chain letters • Sales organizations- pyramids • Computer science -- searching, sorting, coding • Chemistry • Counting

  11. Theorem 1 Theorem 1: An undirected graph is a tree iff there is a unique simple path between any of the vertices. Proof: Assume that there is a unique simple path between any two vertices. Therefore, the graph is ____ To show that it is a tree, show that ____________ To see this, suppose T has a simple circuit containing x and y. There, there would be ________ This is a contradiction. So, we conclude that ___________ …

  12. …proof Theorem 1: An undirected graph is a tree iff there is a unique simple path between any of the vertices. …proof: Assume T is a tree. So T is connected and there exists a path between ________ Let x and y be vertices. To see that there exists a unique simple path, assume not. So, assume that there are 2 paths. Therefore, there is a ________. Thm. 1 from 9.4 says that it is simple. So it is not a tree, and we have a contradiction. In conclusion, ___________

  13. Thm. 2, 3 Theorem 2: A tree with n vertices has n-1 edges Theorem 3: A full m-ary tree with i internal vertices contains n=mi+1 vertices Note: full m-ary tree: n=mi+1, n=l+i

  14. Thm. 4 Theorem 4: A full m-ary tree with n vertices has i=(n-1)/m and l=n – i i internal vertices has n=mi+1 and l=n - i l leaves has i=(l-1)/(m-1) and n=l+i Pf:

  15. Ex Ex: A chain starts when a person sends it to 5 others. These either ignore it or send it to 5 more. If 21 people receive it, how many sent it? Did not? N=21, i= ? l = 17? Ex: 5-ary 2001 see it=n Number who sent it=? Number who didn’t=?

  16. …Ex Ex: 5-ary chain letter 10,000 send it Number who receive it= ? Number who don’t send it = ?

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