23-Introduction to Trees - EECS 210 Discrete Structures...

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Introduction to Trees David O. Johnson EECS 210 (Fall 2016) 1 EECS 210 Discrete Structures David O. Johnson Fall 2016
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Reminders Thanksgiving Holiday, Thursday, November 24 – No Assignment! Homework 6 due: Thursday, December 1 at the beginning of your lecture Introduction to Trees David O. Johnson EECS 210 (Fall 2016) 2
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Any Questions? Introduction to Trees 3 David O. Johnson EECS 210 (Fall 2016)
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Introduction to Trees (Section 11.1) Introduction to Trees Trees as Models Rooted Trees Properties of Trees Introduction to Trees David O. Johnson EECS 210 (Fall 2016) 4
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Introduction In Chapter 10 we showed how graphs can be used to model and solve many problems. In this chapter we will focus on a particular type of graph called a tree, so named because such graphs resemble trees. The undirected graph representing a family tree is an example of a tree. Family trees are graphs that represent genealogical charts. Family trees use vertices to represent the members of a family and edges to represent parent–child relationships. The family tree of the male members of the Bernoulli family of Swiss mathematicians is shown here. Introduction to Trees David O. Johnson EECS 210 (Fall 2016) 5
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Introduction A connected graph that contains no simple circuits (i.e., no path from a vertex through the graph back to the vertex) is called a tree . Trees were used as long ago as 1857, when the English mathematician Arthur Cayley used them to count certain types of chemical compounds. Since that time, trees have been employed to solve problems in a wide variety of disciplines which we will discuss later. Trees are particularly useful in computer science, where they are employed in a wide range of algorithms. Efficient algorithms for locating items in a list Huffman coding - efficient codes for compressing data for transmission and storage AI applications to determine winning strategies for playing games such as checkers and chess Decision trees (basic machine learning technique) - to model procedures carried out using a sequence of decisions Introduction to Trees David O. Johnson EECS 210 (Fall 2016) 6
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Introduction Procedures for building trees containing every vertex of a graph, including depth- first search and breadth-first search, can be used to systematically explore the vertices of a graph. Exploring the vertices of a graph via depth-first search, also known as backtracking, allows for the systematic search for solutions to a wide variety of problems, such as determining how eight queens can be placed on a chessboard so that no queen can attack another. We can assign weights to the edges of a tree to model many problems. For example, using weighted trees we can develop algorithms to construct networks containing the least expensive set of telephone lines linking different network nodes.
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