CMPS 2433
Discrete Structures
Chapter 5 - Trees
R. HALVERSON – MIDWESTERN STATE UNIVERSITY
Basic Tree
Tree: Connected graph with no cycles
Theorem 5.1: Let U & V be vertices in a tree. Then
there is exactly one simple path between U & V.
• Since connected, there exists at least one path
• If 2, then there must be a cycle (discussion pg. 231)
More Tree Theorems
• 5.2: In a tree with more than 1 vertex, there are at
least 2 vertices of degree 1.
• 5.3: A tree with n vertices has exactly n-1 edges.
(Proof by induction – p. 232)
• 5.4a: If an edge is removed from a tree, the
resulting graph is not connected (so not a tree)
• 5.4b: If an edge is added to a tree, the result is a
cycle (so not a tree)
Theorem 5.5:
Following statements are equivalent for a graph T
a)
b)
c)
d)
e)
f)
T is a tree
T is connected & V = E+1
T has 0 cycles & V = E+1
There is exactly 1 simple path between any 2 vertices
T is connected & removal of any edge results in graph that
is not connected
T has no cycles & addition of edge between 2 non-adjacent
vertices results in a cycle
Homework
• Section 5.1 – pages 234-236
• Exercises 1 – 18, 27
Example 5.2 – page 229
• Build telephone network among a group towns
using fewest lines possible; connections do not
have to be direct, just connected
• Consider 5 towns
• Fully connected
• Tree
• See also Example 5.6 – p. 238 – Oil Pipeline
Spanning Tree
• Spanning tree of a graph G is a tree (connected
graph with no cycles) containing all vertices of G
• May not be unique
• Note: A tree is a spanning tree.
• How? Remove 1 edge from each cycle.
• How do you find the cycles?
Breadth First Search Algorithm
• Using from previous chapter
• Edges connecting vertex to its predecessor forms a
tree
• Can start at any vertex
• Produces Shortest Paths Tree
Theorem 5.6
• A graph is connected iff it has a spanning tree.
Minimal & Maximal Spanning Trees
• Applies to Weighted Graphs
• Minimal Spanning Tree
• No other spanning tree of the graph has smaller weight
• May be multiple with same weight
• Figure 5.15 – pg. 242
Minimal Spanning Tree - Prim’s Algorithm
•
•
•
•
Select any vertex, V1
Select smallest edge incident on V1
Now, some V2 is also included in selection
Select smallest edge with 1 end a selected vertex &
other end not selected vertex
• Repeat until all vertices are included
Prim’s Algorithm
• Complexity O(n3) – n = number vertices
• Maximal Spanning Tree?? How can we do this?
Kruskal’s Minimal Spanning Tree Alg. (p. 252)
• Start with all vertices in separate sets
• Select smallest edge; the 2 vertex endpoints have been
selected; Union the sets containing the 2 vertices
• Select smallest edge for which 2 endpoints are NOT in
same set
• Repeat until all vertices are in one set
How is this different from Prim’s?
Homework
• Section 5.2 – Pages 248 – 253
• Exercises 1 – 43 (except 12, 13, 36, 37)
Section 5.3 – Depth-First Search
• OMIT
5.4 Rooted Trees
• Example: Family Tree
• Rooted Tree: a directed graph T such that
1. If ignore direction of edges, graph is a tree
2. There exists unique vertex R with in-degree 0, in-degree
of all other V is 1 (R is called the Root)
• Usually draw with Root at top & other edges going
down
Theorem 5.9
In a rooted tree:
1. V = E + 1
2. No Directed Cycles
3. Unique simple directed path from root to all other
Vertices
Tree Terminology (page 268)
•
•
•
•
•
•
Parent
Child
Ancestor
Descendant
Terminal Vertex – Leaf
Internal Vertex
Section 5.4 Homework
• Omit Pages 269 – 270
• Exercises Pages 271-274
• 1 – 12, 14, 20 – 25, 33 – 38
5.5 Binary Trees & Traversals
• A rooted tree in which each vertex has at most 2
children, denoted left child & right child
• Left subtree of V: subtree rooted by left child of V
• Right subtree
5.6 Binary Search Tree (BST) (p.296)
Consider – each vertex has a value.
BST: For every vertex, all vertices in left subtree have
greater value that root and all vertices in right subtree
have values less than root.
Insertion to a Binary Search Tree (p. 298)
BST Traversals
• Inorder Traversal (p.282)
• Preorder Traversal (p. 277)
• Postorder Traversal (p. 281)
Searching a BST (p. 300)
Homework – 5.5 & 5.6
• 5.5: pages 284+
• Exercises 7 – 30, 55 – 64
• 5.6: pages 302+
• Exercises 48 – 73, demonstrate 74 with 2 different BST’s