21-301: Combinatorics Spring 2019 Lecture 3: Trees Lecturer: John Mackey Date: January 18, 2019 1 Review Hamiltonian Circuit A simple closed path that passes through each vertex in the graph exactly once. A graph is Hamiltonian iff it has a polygon as a spanning subgraph. Consider the two graphs below: The graph on the left is the dodecahedron, which has a spanning polygon and is Hamiltonian. The graph on the right is called the Peterson graph, which does not have a spanning polygon and is not Hamiltonian. 2 Labeled Trees For all following sections, these definitions will be helpful: [n] The set of all positive integers up to and including n . [n] n - 2 The set of sequences of length n - 2, where the numbers in each sequence are chosen from [ n ]. Theorem 2.1. There are n n - 2 different labeled trees on n vertices. To help illustrate this theorem, consider labeling a 3 vertex graph with . The only nonisomorphic tree with 3 vertices is that of a straight line, and we can get a new labeling by switching the label on the center vertex: 1
Lecture 3: Trees 2 There are 3 = 3 3 - 2 ways to do this. Therefore our theorem holds for trees with 3 vertices. We can show that this theorem works for trees with 4 vertices as well. There are two such nonisomorphic trees: There are 12 ways to label the first tree and 4 ways to label the second. Therefore there are 16 = 4 4 - 2 ways to label a tree with 4 vertices. There are three nonisomorphic trees on 5 vertices: There are 60 ways to label the first tree, 5 ways to label the second, and 60 ways to label the third. Therefore there are 125 = 5 5 - 2 ways to label a tree with 5 vertices. 3 Pr¨ufer Codes The proof of Theorem 2.1 relies on an algorithm developed by H. Pr¨ufer that can assign any tree T a unique name in [ n ] n - 2 that characterizes the tree. This is called the tree’s Pr¨ufer code.
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