1 / 16

Applied Combinatorics, 4th Ed. Alan Tucker

Applied Combinatorics, 4th Ed. Alan Tucker. Section 3.1 Properties of Trees Prepared by Joshua Schoenly and Kathleen McNamara. Definitions.

mayten
Download Presentation

Applied Combinatorics, 4th Ed. Alan Tucker

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Applied Combinatorics, 4th Ed.Alan Tucker Section 3.1 Properties of Trees Prepared by Joshua Schoenly and Kathleen McNamara Tucker, Sec. 3.1

  2. Definitions • Tree: a tree is a special type of graph that contains designated vertex called a root so that there is a unique path from the root to any other vertex in the tree. Equivalently, a tree graph contains no circuits. • Rooted Tree: a directed tree graph with all edges directed away from the root a Root = the unique vertex with in-degree of 0 b c e f d i h g j Tucker, Sec. 3.1

  3. Level Number: the length of the path from the root a to x • Parent: the vertex y is a parent of x if they are connected by an edge • Children: the vertex y is a child of x if they are connected by an edge • Siblings: two vertices with the same parent a Parent of g and h b c Level 1 f Level 2 d e Level 3 j g k i h Siblings of each other Children of d Tucker, Sec. 3.1

  4. b a c a b c d f d e f e Theorem 1 A tree with n vertices has n – 1 edges. Proof Choose a root, and direct all edges away from the root. Since each vertex except the root has a single incoming edge, there are n –1 non-root vertices and hence n –1 edges. Root = a Tucker, Sec. 3.1

  5. Definitions • Leaves: vertices with no children • Internal Vertices: vertices with children • m-ary Tree: when each internal vertex of a rooted tree has m children • Binary Tree: when m = 2 Internal vertices Leaves Tucker, Sec. 3.1

  6. Definitions • Height of a Rooted Tree: the length of the longest path to the root. • Balanced Tree (“good”): if all the leaves are at levels h and h-1, where h is the height of the tree. Height (h) is 3 h-1 h Tucker, Sec. 3.1

  7. Theorem 2 If T is an m-ary tree with n vertices, of which i vertices are internal, then, n = mi + 1. Proof Each vertex in a tree, other than the root, is the child of a unique vertex. Each of the i internal vertices has m children, so there are a total of mi children. Adding the root gives n = mi + 1 m = 3 i = 3 n = 10 Tucker, Sec. 3.1

  8. Corollary T is a m-ary tree with n vertices, consisting of i internal vertices and l leaves. Note: The proof of the corollary’s formulas follow directly from n=mi+1(Theorem 2) and the fact that l + i = n Tucker, Sec. 3.1

  9. Example 1 If 56 people sign up for a tennis tournament, how many matches will be played in the tournament? Setting up as a binary tree, there will be 56 leaves and i matches with two entrants entering a match. Entrants Matches Shortened Graph Tucker, Sec. 3.1

  10. Theorem3 • T is a m-ary tree of height h with l leaves. • l≤ mh and if all leaves are at height h, l = mh • h ≥ `élogmlù and if the tree is balanced, h = élogmlù l = 5 h = 3 m = 2 a b c d i f e h g Tucker, Sec. 3.1

  11. Prufer Sequence There exists a sequence (s1, s2,…,sn-2) of length n-2. This is called a Prufer Sequence. Start with the leaf of the smallest label (2). Its neighbor is 5. 5 = s1Delete the edge. Take the next smallest leaf (4). Its neighbor is 3. 3 = s2 Delete the edge. Continue like this obtaining, 1 3 5 7 2 4 6 (5,3,1,7,3,6) 8 Note: There is a 1:1 correspondence to the Prufer Sequence and the tree Tucker, Sec. 3.1

  12. 1 6 3 8 5 7 2 4 Example 2 2, 3, Find the graph that has the Prufer Sequence (6, 2, 3, 3) 1 4 5 2 6 7 3 8 Tucker, Sec. 3.1

  13. Theorem 4 There are nn-2 different undirected trees on n items. 4 8 1 6 3 5 3 6 2 1 7 4 7 5 2 8 Two different trees on 8 items. Tucker, Sec. 3.1

  14. Proof of Theorem 4 There are nn-2 different undirected trees on n items. We showed there is a 1-to-1 correspondence between trees on n items and Prufer sequences of length n-2. Count Prufer sequences. (__, __, __, __, __, __) … n choices n choices This means there are nn-2 different Prufer sequences. Since each tree has a unique Prufer Sequence, there are nn-2 different trees. Tucker, Sec. 3.1

  15. 2 1 5 8 4 6 7 3 Class Problem Create a Prufer Sequence from the graph: Tucker, Sec. 3.1

  16. 2 1 5 4 7 3 Solution Create a Prufer Sequence from the graph: 8 6 (5,6,1,1,5,6) Tucker, Sec. 3.1

More Related