Is My Reasoning Incorrect? A 5-ary Tree with 4n+2 Vertices

• 0rthodontist
In summary, the conversation revolved around determining the number of vertices of degree 5 in a tree with 4n+2 vertices. One person used formulas and arrived at the conclusion that there are n vertices of degree 5. Another person used the fact that a 5-ary tree with k internal vertices has 5k + 1 total vertices, but this was marked wrong. The discussion then shifted to whether or not there is a 5-ary tree with 4n+2 vertices. It was determined that a tree with only vertices of degree 5 or 1 may not necessarily be a 5-ary tree.
0rthodontist
This was a question on the combinatorics midterm: if a tree contains only vertices of degree 5 and degree 1, and the tree has 4n+2 vertices, how many vertices of degree 5 are there?

With k the # of vertices of degree 5 and j the # of vertices of degree 1, the accepted answer used the formulas 5k + j = 8n + 2 (in other words sum over degrees of vertices = 2e) and k + j = 4n + 2 to reach the conclusion k = n.

I used the (fact)? that a 5-ary tree with k internal vertices has 5k + 1 total vertices, to get k = (4n+1)/5. This was marked wrong. I believe that the contradiction between the two answers shows that there is no 5-ary tree with 4n+2 vertices for any n. Is my reasoning incorrect? Is there a 5-ary tree with 4n+2 vertices? Maybe I am confused.

Ah, I got it. A tree with all vertices of degree 5 or 1 is not usually a 5-ary tree.

1. What is a 5-ary tree with 4n+2 vertices?

A 5-ary tree with 4n+2 vertices is a type of tree structure in computer science where each parent node can have up to 5 child nodes, and the total number of vertices in the tree is equal to 4 multiplied by some integer n, plus 2.

2. How is a 5-ary tree with 4n+2 vertices different from other types of trees?

Compared to a binary tree (where each parent node can have up to 2 child nodes) or a ternary tree (where each parent node can have up to 3 child nodes), a 5-ary tree with 4n+2 vertices allows for a larger number of child nodes, making it a more efficient data structure for certain applications.

3. What are some real-world applications of a 5-ary tree with 4n+2 vertices?

A 5-ary tree with 4n+2 vertices can be used for organizing and storing data in databases, representing hierarchical data structures in computer programs, and for optimizing search algorithms.

4. How do you determine if your reasoning about a 5-ary tree with 4n+2 vertices is incorrect?

If you are using a 5-ary tree with 4n+2 vertices in your research or programming and are unsure if your reasoning is correct, you can consult with peers or experts in the field, conduct further research, or test your reasoning with different data sets to validate your conclusions.

5. Can a 5-ary tree with 4n+2 vertices have any number of vertices?

Yes, a 5-ary tree with 4n+2 vertices can have any number of vertices as long as the total number of vertices follows the formula of 4n+2, where n is an integer. This means that the number of vertices can vary depending on the size and structure of the tree.

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