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(a) (i) Give the prefix and postfix notation for the following ordered rooted tree below. (ii) Evaluate the prefix expression found. 9- X- 5

(a) (i) Give the prefix and postfix notation for the following ordered rooted tree below. (ii) Evaluate the prefix expression found. 9- X- 5

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**Educational Website Content** ### Topic: Expression Trees and Graph Theory #### (a) Expression Trees 1. **Problem Statement:** - **(i)** Give the prefix and postfix notation for the following ordered rooted tree. - **(ii)** Evaluate the prefix expression found. 2. **Tree Description:** - The tree is structured with operations and operands: - The root node is a "+" operator. - The left subtree under the root is: - The "*" operator, with its operands: - "÷" operator, with operands 9 and 3. - "+" operator, with operands 4 and the "-" operator with operands 7 and 2. - The right subtree under the root is: - "-" operator, with operands 6 and 5. 3. **Prefix and Postfix Notations:** - **Prefix Notation (Polish notation):** + * ÷ 9 3 + 4 - 7 2 - 6 5 - **Postfix Notation (Reverse Polish notation):** 9 3 ÷ 4 7 2 - + * 6 5 - + 4. **Evaluate the Prefix Expression:** - Break down the expression to solve: - ÷ 9 3 = 3 - - 7 2 = 5 - + 4 5 = 9 - * 3 9 = 27 - - 6 5 = 1 - + 27 1 = 28 - Result: 28 #### (b) Graph Theory **Statement:** - Prove that an undirected graph has an even number of vertices of odd degree. **Explanation:** - In graph theory, the degree of a vertex is the number of edges connected to it. The Handshaking Lemma states that the sum of the degrees of all vertices in an undirected graph is twice the number of edges, thus always even. Therefore, the number of vertices with an odd degree must be even to produce an even sum. By exploring these fundamental concepts, students gain comprehension of both expression evaluation using trees and basic properties in graph theory.
Transcribed Image Text:**Educational Website Content** ### Topic: Expression Trees and Graph Theory #### (a) Expression Trees 1. **Problem Statement:** - **(i)** Give the prefix and postfix notation for the following ordered rooted tree. - **(ii)** Evaluate the prefix expression found. 2. **Tree Description:** - The tree is structured with operations and operands: - The root node is a "+" operator. - The left subtree under the root is: - The "*" operator, with its operands: - "÷" operator, with operands 9 and 3. - "+" operator, with operands 4 and the "-" operator with operands 7 and 2. - The right subtree under the root is: - "-" operator, with operands 6 and 5. 3. **Prefix and Postfix Notations:** - **Prefix Notation (Polish notation):** + * ÷ 9 3 + 4 - 7 2 - 6 5 - **Postfix Notation (Reverse Polish notation):** 9 3 ÷ 4 7 2 - + * 6 5 - + 4. **Evaluate the Prefix Expression:** - Break down the expression to solve: - ÷ 9 3 = 3 - - 7 2 = 5 - + 4 5 = 9 - * 3 9 = 27 - - 6 5 = 1 - + 27 1 = 28 - Result: 28 #### (b) Graph Theory **Statement:** - Prove that an undirected graph has an even number of vertices of odd degree. **Explanation:** - In graph theory, the degree of a vertex is the number of edges connected to it. The Handshaking Lemma states that the sum of the degrees of all vertices in an undirected graph is twice the number of edges, thus always even. Therefore, the number of vertices with an odd degree must be even to produce an even sum. By exploring these fundamental concepts, students gain comprehension of both expression evaluation using trees and basic properties in graph theory.
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