1. Warm Up – Wednesday

2. Trees A tree is a network without any circuits.
Note: Our definition of circuit here is different than an Euler Circuit or a Hamilton Circuit.
The graph to the left has a general circuit starting
at C and following C, B, E, D, C. A circuit simply starts
and ends at the same vertex.

3. Not a tree!
Because this graph has a circuit, this network is not a tree!

4. The following are trees!
They have no circuits within the graph.

5. Key Properties of Trees
The single path property: In a tree, there is only one path connecting two vertices.
Notice if we have two paths then we also have a circuit and thus a non-tree.

6. Key Properties of Trees
All Bridges Property: In a tree, every edge is a bridge (Remember that a bridge is an edge that if removed, makes the graph disconnected).
The N-1 edges property: A tree with N vertices has N-1 Edges.

7. Key Properties of Trees
If a network satisfies one of those three properties then it satisfies all three, and is a tree! WOOT!
The single path property: In a tree, there is only one path connecting two vertices.
All Bridges Property: In a tree, every edge is a bridge (Remember that a bridge is an edge that if removed, makes the graph disconnected).
The N-1 edges property: A tree with N vertices has N-1 Edges.

8. Trees #1

9. Subtrees A subtree is a tree within a network.
Consider the network above. Do you see any trees within
the graph (meaning if I ignore certain edges)?

10. Spanning Trees
A spanning tree is a subtree that uses all the vertices.

11. Spanning Trees Thus we have three different spanning trees!
If I remove one of the three edges in the triangular path, I get a tree because I have removed the circuit.
Thus we have three different spanning trees!

12. Redundancy
The redundancy is a number that helps us when counting spanning trees in a graph.
The redundancy is the number of Edges (M) minus one less than the number of vertices (N – 1).
𝑹=𝑴− 𝑵−𝟏
We define redundancy this way so a tree has a redundancy of 0!

13. Redundancy
The redundancy tells you the number of edges I need to remove in order to find a spanning tree.
I cannot remove Bridges! (Edges that would disconnect the graph)

14. Example #1
Calculate the Redundancy and find a spanning tree in the graph below.

15. Example #1 There are 12 vertices and 14 edges. Thus my redundancy is:
14 – 12 – 1 =3
This means I need to remove 3 edges to get a spanning tree.

16. Example #1
If I remove edges BE, HI, and IJ, a spanning tree is created.
Notice we still have a network and there are no circuits.