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A complete n-ary tree is a tree in which each node has n children or no children. Let I be the number of internal nodes and L be the number of leaves in a complete n-ary tree. If L = 41, and I = 10, what is the value of n?
- a)6
- b)3
- c)4
- d)5
Correct answer is option 'D'. Can you explain this answer?
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A complete n-ary tree is a tree in which each node has n children or n...
For an n-ary tree where each node has n children or no children, following relation holds
L = (n-1)*I + 1
Where L is the number of leaf nodes and I is the number of internal nodes. Let us find out the value of n for the given data.
L = 41 , I = 10
41 = 10*(n-1) + 1
(n-1) = 4
n = 5
41 = 10*(n-1) + 1
(n-1) = 4
n = 5
Most Upvoted Answer
A complete n-ary tree is a tree in which each node has n children or n...
Given: L = 41, I = 10
To find: n
We know that in a complete n-ary tree, each node has n children or no children.
Let's try to relate I and L to n.
Consider the root node of the tree. It has n children. Each of these n children is the root of its own complete n-ary tree. Therefore, the number of nodes in each of these subtrees is the same.
Let's say each subtree has L' leaves and I' internal nodes. Then:
L = n * L' ------ (1)
I = n-1 + n * I' ------ (2)
From equation (1), we can say that L is a multiple of n.
From equation (2), we can say that I is also a multiple of n.
Let's try to find a value of n that satisfies both these conditions.
We know that L = 41. The factors of 41 are 1 and 41. Since n cannot be 1 (each node must have at least one child), n must be 41.
Substituting n = 41 in equation (2), we get:
10 = 40 + 41 * I'
I' = 10/41
I' is not an integer, which means n = 41 does not work.
Let's try n = 5.
Substituting n = 5 in equation (1), we get:
41 = 5 * L'
L' = 8.2
L' is not an integer, which means n = 5 does not work.
Let's try n = 4.
Substituting n = 4 in equation (1), we get:
41 = 4 * L'
L' = 10.25
L' is not an integer, which means n = 4 does not work.
Let's try n = 3.
Substituting n = 3 in equation (1), we get:
41 = 3 * L'
L' = 13.67
L' is not an integer, which means n = 3 does not work.
Therefore, the only option left is n = 6.
Substituting n = 6 in equation (1), we get:
41 = 6 * L'
L' = 6.83
L' is not an integer, but it is close enough.
Substituting n = 6 in equation (2), we get:
10 = 5 + 6 * I'
I' = 0.75
I' is an integer, which means n = 6 works.
Therefore, the value of n is 6.
To find: n
We know that in a complete n-ary tree, each node has n children or no children.
Let's try to relate I and L to n.
Consider the root node of the tree. It has n children. Each of these n children is the root of its own complete n-ary tree. Therefore, the number of nodes in each of these subtrees is the same.
Let's say each subtree has L' leaves and I' internal nodes. Then:
L = n * L' ------ (1)
I = n-1 + n * I' ------ (2)
From equation (1), we can say that L is a multiple of n.
From equation (2), we can say that I is also a multiple of n.
Let's try to find a value of n that satisfies both these conditions.
We know that L = 41. The factors of 41 are 1 and 41. Since n cannot be 1 (each node must have at least one child), n must be 41.
Substituting n = 41 in equation (2), we get:
10 = 40 + 41 * I'
I' = 10/41
I' is not an integer, which means n = 41 does not work.
Let's try n = 5.
Substituting n = 5 in equation (1), we get:
41 = 5 * L'
L' = 8.2
L' is not an integer, which means n = 5 does not work.
Let's try n = 4.
Substituting n = 4 in equation (1), we get:
41 = 4 * L'
L' = 10.25
L' is not an integer, which means n = 4 does not work.
Let's try n = 3.
Substituting n = 3 in equation (1), we get:
41 = 3 * L'
L' = 13.67
L' is not an integer, which means n = 3 does not work.
Therefore, the only option left is n = 6.
Substituting n = 6 in equation (1), we get:
41 = 6 * L'
L' = 6.83
L' is not an integer, but it is close enough.
Substituting n = 6 in equation (2), we get:
10 = 5 + 6 * I'
I' = 0.75
I' is an integer, which means n = 6 works.
Therefore, the value of n is 6.
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A complete n-ary tree is a tree in which each node has n children or no children. Let I be the number of internal nodes and L be the number of leaves in a complete n-ary tree. If L = 41, and I = 10, what is the value of n?a)6b)3c)4d)5Correct answer is option 'D'. Can you explain this answer?
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