1. Chap 8 Trees Def 1: A tree is a connected,undirected, graph with no simple circuits. Ex1. Theorem1: An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices.

2. Chap 8 Trees Root, rooted tree, parent, child, siblings, ancestors, descendants,leaf,subtree,internal vertices:have children Ex2 Def 2: m-ary tree : every internal vertex has no more than m children;full m-ary tree ; binary tree.

3. Chap 8 Trees Trees as models Properties of trees Theorem2: A tree with n vertices has n-1 edges. Theorem3: A full m-ary tree with i internal vertices contains n=mi+1 vertices.

4. Chap 8 Trees Level of a vertex :length of the unique path from the root to the vertex height of a rooted tree:length of the longest path from the root Ex10 a rooted m-ary tree of height h is balance if all leaves are at levels h or h-1

5. Chap 8 Trees Tree traversal ordered rooted tree ; left/right child/subtree Def1: preorder traversal of an ordered rooted tree Fig 2; Ex2 Def 2: inorder traversal Fig 5, Ex3 Def 3: postorder traversal Fig 7, Ex4

6. Chap 8 Trees Fig 9 : preorder : list each vertex the first time this curve passes it. inorder : list a leaf the first time the curve passes it ; list each internal vertex the second time the curve passes it postorder :list a vertex the last time it is passed on the way back up to its parent.

7. Chap 8 Trees Represent complicated expressions using ordered rooted trees Ex5 inorder traversal produces the original expression with the elements and operations in the same order as they originally occurred. infix form ( Fig 11) : need to include parentheses whenever an operation is encountered in the inorder traversal

8. Chap 8 Trees Prefix form : no parenthesis are needed (Polish notation) Ex6 We can evaluate an expression in prefix form by working from right to left Ex7 Postfix form (reverse Polish notation) : no parenthesis are needed

9. Chap 8 Trees Ex8 evaluate an expression from left to right Ex9 Ex10 Because prefix and postfix expressions are unambiguous and can easily be evaluated, they are extensively used in computers science.

10. Chap 8 Trees Spanning Tree Def1 : let G be a simple graph.A spanning tree of G is a subgraph of G that is a tree containing every vertex of G (n vertices with n-1 edges) Ex1. Algorithm for constructing spanning tree depth-first search /backtracking Ex3. breadth-first search Ex4.

11. Chap 8 Trees Minimum Spanning Tree Def 1 : A minimum spanning tree in a connected weighted graph is a spanning tree that has the smallest possible sum of weights of its edges Prim’s algorithm Ex2. Kruskal’s algorithm Ex3.