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Lecture 17 Trees

Lecture 17 Trees. CSCI – 1900 Mathematics for Computer Science Spring 2014 Bill Pine. Lecture Introduction. Reading Kolman Section 7.1, 7.2 Review Graphs Trees Rooted Trees Specialized Trees. 3. 1. 2. Review of Directed Graphs.

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Lecture 17 Trees

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  1. Lecture 17Trees CSCI – 1900 Mathematics for Computer Science Spring 2014 Bill Pine

  2. Lecture Introduction • Reading • Kolman Section 7.1, 7.2 • Review Graphs • Trees • Rooted Trees • Specialized Trees CSCI 1900

  3. 3 1 2 Review of Directed Graphs • A Digraph, G=(V, A), consists of a finite set V of objects called vertices, and a finite set A of objects called arcs between the vertices • Arcs have a direction • Example: V={1, 2, 3} A= { (1, 2), (3, 2) } CSCI 1900

  4. 3 1 2 Graphs • A graph, G=(V, E), consists of a finite set V of objects called vertices, and a finite set E of objects called edges • Edges have no direction • Example: V={1,2,3} E= { (1,2), (2,3) } CSCI 1900

  5. Connected Graph • A graph is connected if there is a path (sequence of edges) between any two pairs of vertices Connected Disconnected CSCI 1900

  6. C4 C3 Cycle • If you can start at a node and move along edges and come back to the same node the graph has a cycle CSCI 1900

  7. Not an acyclic graph A disconnected acyclic graph Acyclic Graphs • A graph is acyclic if it has no cycles CSCI 1900

  8. Trees • Let A be a set and let T be a relation on A. We say that T is a tree if there is a vertex v0 with the property that there exists a unique path in T from v0 to every other vertex in A, but no path from v0 to v0 -Kolman, Busby, and Ross, p. 271 Or • A tree is a connected acyclic graph CSCI 1900

  9. Rooted Trees • In computer science, we work primarily with rooted trees • A rooted tree is a directed graph • Which has one vertex, v0 , called the root, with no arcs coming into it • In-degree of v0 = 0 • Any other vertex, vi, in the tree has exactly one path to it from v0 • In-degree of vi = 1 • There may be any number of paths from any vertex CSCI 1900

  10. Examples of Trees • From this point forward, we will follow the computer science conventions • Tree means rooted tree • Edge means directed edge • Using the definitions given, determine which of the following examples are trees and which are not CSCI 1900

  11. Is this a tree? CSCI 1900

  12. Is this a tree? CSCI 1900

  13. CSCI 1100 Using Info Technology CSCI 1710 WWW Design CSCI 1510 Student in University CSCI 1800 Visual Prog I CSCI 2150 Computer Organization CSCI 2800 Visual Prog Adv Concepts CSCI 2235 Intro to Unix CSCI 3400 Network Fundamentals CSCI 2910 Client & Server Side Prog CSCI 4417 System Administration CSCI 3220 Intro to Database CSCI 3250 Software Engineering I CSCI 4227 Advanced Database CSCI 4800 Senior Capstone Technology CSCI 3350 Software Engineering II CSCI 4217 Ethical Issues Is this a tree? CSCI 1900

  14. Level 0 Level 1 Level 2 Level 3 Definitions Level – all of the vertices located n-edges from v0 are said to be at level n CSCI 1900

  15. More Definitions • A vertex, v, is considered the parent of all of the vertices connected to it by edges leaving v • A vertex, v, is considered the child of the vertex connected to the single edge entering v • A vertex, v, is considered the sibling of all vertices at the same level with the same parent CSCI 1900

  16. More Definitions • A vertex vj is considered a descendant of a vertex vi if there is a path from vi to vj • The height of a tree is the number of the largest level • The vertices of a tree that have no children are called leaves CSCI 1900

  17. v0 v4 v1 v2 v3 v8 v5 v6 v7 Tree Example Level 0 • v0 is the parent of v1 , v2 , v3 , and v4 • v1 , v2 , v3 , and v4 are children of v0 • v1 , v2 , v3 , and v4 are siblings • v5 , v6 , v2 , v7 , v8 , and v4 are leaves (no children) • Height of the tree is 2 Level 1 Level 2 CSCI 1900

  18. More Definitions • If every vertex of a tree has at most n children, then the tree is considered an n-tree • If every vertex of a tree with offspring has exactly n offspring, then the tree is considered a complete n-tree • When n=2, the tree is a binary tree CSCI 1900

  19. Examples • If the set A = {a, b, c, d, e} represents all of the vertices for a tree T, what is the maximum possible height of T? What is the minimum possible height of T? • If the set A = {a, b, c, d, e} represents all of the vertices for a tree T and T is a complete binary tree, what is the maximum height of T? CSCI 1900

  20. Examples (cont) • If the height of a complete 4-tree is 3, how many leaves does this tree have? CSCI 1900

  21. Key Concepts Summary • Review Graphs • Trees • Rooted Trees • Specialized Trees • Reading for next time • Kolman Section 7.2, 7.3 CSCI 1900

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