CS103
HO#34
Slides--Intorduction to Finite Automata
April 30, 2010
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CS103
4/30/10
Midterm Exam
Review Sessions:
Saturday, 4 – 7pm, Gates B12
Sunday, 4 – 7pm, Gates B12
(bring ID for entry)
Exam: Tuesday, May 4, 7:00 – 9:00 pm
Locations to be announced
The exam is open book, open notes.
You may not use computers or mobile devices.
Mathematical Foundations of Computing
Proving Things About Trees
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Theorem:
If T is a tree and x is one of its nodes other than the root,
then the graph X consisting of x, its descendants, and the edges
connecting them is a tree.
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Proving Things About Trees
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Theorem:
If T is a tree and x is one of its nodes other than the root,
then the graph X consisting of x, its descendants, and the edges
connecting them is a tree.
Proof:
There is only one path from the root to x, so removing the edge
in that path that is incident on x disconnects X from the original graph.
If there is a cycle in X, then T is not a tree, and if X is not connected
then T is not a tree.
So X is acyclic and connected and thus a tree.
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Proving Things About Trees
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Theorem:
A tree with n nodes has n – 1 edges.
Proof:
Every node in the tree except the root has a unique parent,
and is connected to its parent by an edge.
Every edge connects some node to its parent.
There are n – 1 nodes with parents, so there are n – 1 edges.
Alternate phrasings:
|E| = |V| – 1
|V| = |E| + 1
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Properties of Trees
Let G = (V, E) be an undirected graph.
The following statements are equivalent:
1. G is a tree.
2. Any two vertices of G are connected by a unique simple path.
3. G is connected, but if any edge in E is removed, the result is not connected.
4. G is connected, and |E| = |V| – 1.
5. G is acyclic, and
|E| = |V| – 1.
6. G is acyclic, but if an edge is added to E, the resulting graph has a cycle.
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Theorem:
A tree with n nodes has n – 1 edges.
Base Case:
A tree with one node has no edges.
Inductive Step:
Assume that all trees with k
1 nodes have k – 1 edges,
and consider a tree T with k + 1 nodes.
Show that T has k edges.
T must have a leaf since it is finite, connected, and acyclic.
Let x be a leaf of T, and remove x and the edge connecting it to its parent.
The remaining graph is still a tree and has k nodes.
Thus by the I.H., it
has k – 1 edges, and T has k edges.
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Using Induction To Prove Things About Trees