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5. Let S be the set of full binary trees, defined recursively as follows: • Basic step: a single vertex v with no edges is a full binary tree T,- • Recursive step: if T, and T, are full binary trees, then a new full binary tree T' can be constructed by taking T1 and T2, adding a new vertex v, and adding edges between v and the roots of T, and T2. Prove that n(T) is odd for any full binary tree T, where n(T) is the number of vertices of T.

5. Let S be the set of full binary trees, defined recursively as follows: • Basic step: a single vertex v with no edges is a full binary tree T,- • Recursive step: if T, and T, are full binary trees, then a new full binary tree T' can be constructed by taking T1 and T2, adding a new vertex v, and adding edges between v and the roots of T, and T2. Prove that n(T) is odd for any full binary tree T, where n(T) is the number of vertices of T.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Topic 6: Full Binary Trees** Let \( S \) be the set of full binary trees, defined recursively as follows: - **Basic Step:** A single vertex \( v \) with no edges is a full binary tree \( T_0 \). - **Recursive Step:** If \( T_1 \) and \( T_2 \) are full binary trees, then a new full binary tree \( T' \) can be constructed by taking \( T_1 \) and \( T_2 \), adding a new vertex \( v \), and adding edges between \( v \) and the roots of \( T_1 \) and \( T_2 \). **Objective:** Prove that \( n(T) \) is odd for any full binary tree \( T \), where \( n(T) \) is the number of vertices of \( T \). This structure exemplifies the recursive nature of full binary trees and sets the stage for understanding their properties, particularly focusing on the parity of the vertex count.
Transcribed Image Text:**Topic 6: Full Binary Trees** Let \( S \) be the set of full binary trees, defined recursively as follows: - **Basic Step:** A single vertex \( v \) with no edges is a full binary tree \( T_0 \). - **Recursive Step:** If \( T_1 \) and \( T_2 \) are full binary trees, then a new full binary tree \( T' \) can be constructed by taking \( T_1 \) and \( T_2 \), adding a new vertex \( v \), and adding edges between \( v \) and the roots of \( T_1 \) and \( T_2 \). **Objective:** Prove that \( n(T) \) is odd for any full binary tree \( T \), where \( n(T) \) is the number of vertices of \( T \). This structure exemplifies the recursive nature of full binary trees and sets the stage for understanding their properties, particularly focusing on the parity of the vertex count.
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