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The Fascinating World of Graph Theory$
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Ping Zhang, Gary Chartrand, and Arthur Benjamin

Print publication date: 2017

Print ISBN-13: 9780691175638

Published to Princeton Scholarship Online: May 2018

DOI: 10.23943/princeton/9780691175638.001.0001

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Constructing Trees

Constructing Trees

(p.67) 4 Constructing Trees
The Fascinating World of Graph Theory

Arthur Benjamin

Gary Chartrand

Ping Zhang

Princeton University Press

This chapter considers a class of graphs called trees and their construction. Trees are connected graphs containing no cycles. When dealing with trees, a vertex of degree 1 is called a leaf rather than an end-vertex. The chapter first provides an overview of trees and their leaves, along with the relevant theorems, before discussing a tree-counting problem, introduced by British mathematician Arthur Cayley, involving saturated hydrocarbons. It shows that counting the number of saturated hydrocarbons is the same as counting the number of certain kinds of nonisomorphic trees. It then revisits another Cayley problem, one that involved counting labeled trees, and describes Cayley's Tree Formula and the corresponding proof known as the Prüfer code. It also explores decision trees and concludes by looking at the Minimum Spanning Tree Problem and its solution, Kruskal's Algorithm.

Keywords:   graph, tree, connected graph, leaf, Arthur Cayley, Cayley's Tree Formula, Prüfer code, decision tree, Minimum Spanning Tree Problem, Kruskal's Algorithm

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