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Let T be the tree with Prüfer code 3, 2, 1, 3, 2, 1. (a) Give the number of vertices of T, and the number of vertices that are leaves. Justify your answers. (b) Does T contain a 3-4-walk of length 2? Justify your answer. (c) What is the number of distinct trees with vertex set V(T)? Justify your answer. (d) What is the number of distinct graphs that have T as a spanning tree? Justify your answer. (e) Give an efficient algorithm that, for any sequence d1, d2, ..., dn that is the degree sequence of a tree, constructs a particular tree with that degree sequence. Explain briefly why the algorithm is correct and efficient.

Let T be the tree with Prüfer code 3, 2, 1, 3, 2, 1. (a) Give the number of vertices of T, and the number of vertices that are leaves. Justify your answers. (b) Does T contain a 3-4-walk of length 2? Justify your answer. (c) What is the number of distinct trees with vertex set V(T)? Justify your answer. (d) What is the number of distinct graphs that have T as a spanning tree? Justify your answer. (e) Give an efficient algorithm that, for any sequence d1, d2, ..., dn that is the degree sequence of a tree, constructs a particular tree with that degree sequence. Explain briefly why the algorithm is correct and efficient.

Holt Mcdougal Larson Pre-algebra: Student Edition 2012
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter12: Angle Relationships And Transformations
Section12.5: Reflections And Symmetry
Problem 20E
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Let T be the tree with Prüfer code 3, 2, 1, 3, 2, 1. (a) Give the number of vertices of T, and the number of vertices that are leaves. Justify your answers. (b) Does T contain a 3-4-walk of length 2? Justify your answer. (c) What is the number of distinct trees with vertex set V(T)? Justify your answer. (d) What is the number of distinct graphs that have T as a spanning tree? Justify your answer. (e) Give an efficient algorithm that, for any sequence d1, d2, ..., dn that is the degree sequence of a tree, constructs a particular tree with that degree sequence. Explain briefly why the algorithm is correct and efficient.
Transcribed Image Text:Let T be the tree with Prüfer code 3, 2, 1, 3, 2, 1. (a) Give the number of vertices of T, and the number of vertices that are leaves. Justify your answers. (b) Does T contain a 3-4-walk of length 2? Justify your answer. (c) What is the number of distinct trees with vertex set V(T)? Justify your answer. (d) What is the number of distinct graphs that have T as a spanning tree? Justify your answer. (e) Give an efficient algorithm that, for any sequence d1, d2, ..., dn that is the degree sequence of a tree, constructs a particular tree with that degree sequence. Explain briefly why the algorithm is correct and efficient.
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