1. Graphs and Trees
This handout:
Trees
Minimum Spanning Tree Problem

2. Terminology of Graphs: Cycles, Connectivity and Trees
A path that begins and ends at the same node is called a cycle.
Example:
Two nodes are connected if there is a path between them.
A graph is connected if every pair of its nodes is connected.
A graph is acyclic if it doesn’t have any cycle.
A graph is called a tree if it is connected and acyclic.

3. Examples/Applications of Trees
Family tree
Evolutionary tree
Possibility tree
Prime number factorization tree
Spanning tree of a telecom network
(considered later in this handout)
Hydrocarbon molecules

4. Properties of Trees Definition: Let T be a tree.
If T has at least 3 nodes, then
A node of degree 1 in T is called a leaf (or a terminal node).
A node of degree greater than 1 is called an internal node.
Lemma: Every tree with more than one node
has at least one leaf.
Theorem: For any positive integer n,
any tree with n nodes has n-1 edges.
Proof by induction (blackboard).

5. Properties of Trees Lemma: If G is any connected graph,
C is a cycle in G,
and one of the edges of C is removed from G,
then the graph that remains is still connected.
Theorem: For any positive integer n,
if G is a connected graph with n vertices and n-1 edges, then G is a tree.
if G is an acyclic graph with n vertices and n-1 edges,
then G is a tree.
Proofs on blackboard.

6. Rooted Trees
Definition: A rooted tree is a tree in which one vertex is distinguished from the others and is called the root.
The level of a vertex is the number of edges along the unique path between it and the root.
The height of a rooted tree is the maximum level to any vertex of the tree.
The children of a vertex v are those vertices that are adjacent to v and one level farther away from the root than v.
When every vertex in a rooted tree has at most two children, the tree is called a binary tree. r a b c d

7. Minimum Spanning Tree Problem
Given: Graph G=(V, E), |V|=n
Cost function c: E  R .
Goal: Find a minimum-cost spanning tree for V
i.e., find a subset of arcs E*  E which
connects any two nodes of V
with minimum possible cost.
Example:
Min. span. tree: G*=(V,E*)
G=(V,E) 2 3 4 5 7 8 2 3 4 5 7 8 e b c d a e b c d a
Red bold arcs are in E*

8. Algorithm for solving the Minimum Spanning Tree Problem
Initialization: Select any node arbitrarily,
connect to its nearest node.
Repeat
Identify the unconnected node
which is closest to a connected node
Connect these two nodes
Until all nodes are connected
Note: Ties for the closest node are broken arbitrarily.

9. The algorithm applied to our example
Initialization: Select node a to start.
Its closest node is node b. Connect nodes a and b.
Iteration 1: There are two unconnected node closest to a connected node: nodes c and d
(both are 3 units far from node b).
Break the tie arbitrarily by
connecting node c to node b. 2 3 4 5 7 8 e b c d a
Red bold arcs are in E*;
thin arcs represent potential links. 2 3 4 5 7 8 e b c d a

10. The algorithm applied to our example
Iteration 2: The unconnected node closest to a connected node is node d (3 far from node b). Connect nodes b and d.
Iteration 3: The only unconnected node left is node e. Its closest connected node is node c
(distance between c and e is 4).
Connect node e to node c.
All nodes are connected. The bold
arcs give a min. spanning tree. 2 3 4 5 7 8 e b c d a 2 3 4 5 7 8 e b c d a