March 20/2003 Jayakanth Srinivasan,

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2 Definition : A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges. Definition : In a multigraph G = (V, E) two or more edges may connect the same pair of vertices.

3 Definition : In a pseudograph G = (V, E) two or more edges may connect the same pair of vertices, and in addition, an edge may connect a vertex to itself. pseudographs multigraphs simple graphs

4 Definition : In a directed graph G = (V, E) the edges are ordered pairs of (not necessarily distinct) vertices. Definition : In a directed multigraph G = (V, E) the edges are ordered pairs of (not necessarily distinct) vertices, and in addition there may be multiple edges.

5 TYPE EDGES MULTIPLE EDGES LOOPS ALLOWED? ALLOWED? Simple graph Undirected NO NO Multigraph Undirected YES NO Pseudograph Undirected YES YES Directed graph Directed NO YES Directed multigraph Directed YES YES

6 Definition : Two vertices, u and v in an undirected graph G are called adjacent (or neighbors) in G, if {u, v} is an edge of G. Definition : The degree of a vertex in an undirected graph is the number of edges incident with it, except that a loop at a vertex contributes twice to the degree of that vertex.

7 Theorem : Let G = (V, E) be an undirected graph G with e edges. Then deg( v ) = 2 e v ε V The sum of the degrees over all the vertices equals twice the number of edges Definition : A subgraph of a graph G = (V, E) is a graph H = (W, F) where W V and F E.

8 Definition : A tree is a connected undirected graph with no simple circuits. Theorem : An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices.

9 a) b) c)

10 A vertex that has children is called an internal vertex. The subtree at vertex v is the subgraph of the tree consisting of vertex v and its descendants and all edges incident to those descendants.

11 Definition: A rooted tree is called a binary tree if every internal vertex has no more than 2 children. The tree is called a full binary tree if every internal vertex has exactly 2 children.

12 Definition: An ordered rooted tree is a rooted tree where the children of each internal vertex are ordered. In an ordered binary tree, the two possible children of a vertex are called the left child and the right child, if they exist.

13 Theorem: A tree with N vertices has N-1 edges. Theorem: There are at most 2 H leaves in a binary tree of height H. Corallary: If a binary tree with L leaves is full and balanced, then its height is H = log 2 L.

14 The parent of a non-root vertex is the unique vertex u with a directed edge from u to v. A vertex is called a leaf if it has no children. The ancestors of a non-root vertex are all the vertices in the path from root to this vertex. The descendants of vertex v are all the vertices that have v as an ancestor.

15 The level of vertex v in a rooted tree is the length of the unique path from the root to v. The height of a rooted tree is the maximum of the levels of its vertices. A rooted binary tree of height H is called balanced if all its leaves are at levels H or H-1.

16 Each vertex contains a distinct key value, The key values in the tree can be compared using greater than and less than, and The key value of each vertex in the tree is less than every key value in its right subtree, and greater than every key value in its left subtree.

17 !" # A traversal algorithm is a procedure for systematically visiting every vertex of an ordered binary tree. Tree traversals are defined recursively. Three traversals are named: preorder inorder postorder

18 $%$&%$!" # Let T be an ordered binary tree with root r. If T has only r then r is the preorder traversal. Else let T 1, T 2 be the left and right subtrees at r. Visit R. Traverse T 1 in preorder Traverse T 2 in preorder.

19 '$&%$!" # Let T be an ordered binary tree with root r. If T has only r then r is the inorder traversal. Else let T 1, T 2 be the left and right subtrees at r. Traverse T 1 in inorder Visit R. Traverse T 2 in inorder.

20 $&%$!" # Let T be an ordered binary tree with root r. If T has only r then r is the postorder traversal. Else let T 1, T 2 be the left and right subtrees at r. Traverse T 1 in postorder Traverse T 2 in postorder Visit R.

21 % ROOT INORDER TRAVERSAL: 8-5 has value 3 PREORDER TRAVERSAL: POSTORDER TRAVERSAL: 8 5 -

22 % A special kind of binary tree in which: Each leaf node contains a single operand Each nonleaf node contains a single binary operator The left and right subtrees of an operator node represent subexpressions that must be evaluated before applying the operator at the root of the subtree.

23 % * What value does it have? ( ) * 3 = 18

24 "% * Infix: ( ( ) * 3 ) Prefix: * evaluate from right Postfix: * evaluate from left

25 (! When a binary expression tree is used to represent an expression, the levels of the nodes in the tree indicate their relative precedence of evaluation. Operations at higher levels of the tree are evaluated later than those below them. The operation at the root is always the last operation performed.

26 % * - / Infix: ( ( 8-5 ) * ( ( ) / 3 ) ) Prefix: * / evaluate from right Postfix: / * evaluate from left

Introduction to Computers and Programming. Concept Question

Introduction to Computers and Programming Prof. I. K. Lundqvist Lecture 7 April 2 2004 Concept Question G1(V1,E1) A graph G(V, where E) is V1 a finite = {}, nonempty E1 = {} set of G2(V2,E2) vertices and