1. Aim: Graph Theory - Trees Course: Math Literacy Do Now: Aim: What’s a tree?

2. Aim: Graph Theory - Trees Course: Math Literacy A Tree Tree – a graph with the smallest number of edges that allow all vertices to be reached from all other vertices. all connected no circuits no longer trees

3. Aim: Graph Theory - Trees Course: Math Literacy What’s in a tree? A tree is a graph that is connected and has no circuits. All trees have the following properties: a.There is one and only one path joining any two vertices. b.Every edge is a bridge. c.A tree with n vertices must have n – 1 edges. a tree with 5 vertices must have 4 edges.

4. Aim: Graph Theory - Trees Course: Math Literacy Model Problem Which graph is a tree?

5. Aim: Graph Theory - Trees Course: Math Literacy Spanning Trees Spanning tree – a subgraph of a connected graph that contains no circuits. not a tree 7 vertices 7 edges a tree with removal of BC 7 vertices, 6 edges a tree with removal of EG 7 vertices, 6 edges

6. Aim: Graph Theory - Trees Course: Math Literacy Model Problem Find a spanning tree for the graph below. 8 vertices 12 edges 5 gotta go 8 vertices, 7 edges - a tree

7. Aim: Graph Theory - Trees Course: Math Literacy Model Problem Find a spanning tree for the graph below. 6 vertices 8 edges 3 gotta go

8. Aim: Graph Theory - Trees Course: Math Literacy Efficiency! Minimum spanning tree - a spanning tree with the smallest possible total weight on a weight graph. 8 20 35 1715 12 24 original weighted graph = 131 8 20 35 1715 24 35+24+20+8 +17+15 = 119 8 20 35 1715 12 35+17+12+15+20+8 = 107

9. Aim: Graph Theory - Trees Course: Math Literacy Kruskal’s Algorithm Kruskal’s Algorithm: finding the minimum spanning tree from a weighted graph: 1.Find the edge with the smallest weight in the graph. If there is more than one, pick one at random and mark it. 2.Find the next smallest edge in the graph. If there is more than one, pick at random. Mark it. 3.Find the next-smallest unmarked edge that does not create a red circuit. 4.Repeat step 3 until all vertices are included. The marked edges are the desired minimum spanning tree.

10. Aim: Graph Theory - Trees Course: Math Literacy Model Problem Seven building on a college campus are connected by the sidewalks show in the figure below. The weight graph represents building as vertices sidewalks as edges and sidewalk lengths as weights. A heavy snow has fallen and the sidewalks need to be cleared quickly. Determine the shortest series of sidewalks to clear. What is the total length of the sidewalks that need to be cleared? 264’ 256’ 262’ 242’ 259’255’ 251’ 253’ 251’241’ 245’ 274’

11. Aim: Graph Theory - Trees Course: Math Literacy Model Problem 264’ 256’ 262’ 242’ 259’255’ 251’ 253’ 251’241’ 245’ 274’ G E D B A C F

12. Aim: Graph Theory - Trees Course: Math Literacy Model Problem Kruskal’s Algorithm: finding the minimum spanning tree from a weighted graph: 1.Find the edge with the smallest weight in the graph. If there is more than one, pick one at random and mark it. 2.Find the next smallest edge in the graph. If there is more than one, pick at random. Mark it. 3.Find the next-smallest unmarked edge that does not create a red circuit. 4.Repeat step 3 until all vertices are included. The marked edges are the desired minimum spanning tree. +++++= 1499’

13. Aim: Graph Theory - Trees Course: Math Literacy Model Problem Use Kruskal’s Algorithm to find the minimum spanning tree for the graph below. Give the total weight of the minimum spanning tree.

14. Aim: Graph Theory - Trees Course: Math Literacy The Product Rule

15. Aim: Graph Theory - Trees Course: Math Literacy The Product Rule

16. Aim: Graph Theory - Trees Course: Math Literacy The Product Rule