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In a binary tree with n nodes, every node has an odd number of descendants. Every node is considered to be its own descendant. What is the number of nodes in the tree that have exactly one child?
- a)0
- b)1
- c)(n-1)/2
- d)n-1
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
In a binary tree with n nodes, every node has an odd number of descend...
The given binary tree has n nodes and every node has an odd number of descendants. We need to find the number of nodes in the tree that have exactly one child.
To solve this problem, let's consider some properties of binary trees and the given conditions:
1. Binary Tree: A binary tree is a tree data structure in which each node has at most two children, referred to as the left child and the right child.
2. Odd Number of Descendants: Every node in the given binary tree has an odd number of descendants, which means that the total number of nodes in the subtree rooted at any node is always odd.
3. Every Node is Its Own Descendant: The given condition states that every node is considered to be its own descendant. This means that we need to include each node itself while counting the number of descendants.
Now, let's analyze the possible cases for nodes with exactly one child:
Case 1: Leaf Nodes
- Leaf nodes are the nodes that do not have any children.
- Since every node is considered to be its own descendant, leaf nodes have exactly one child (itself).
- However, leaf nodes do not have any children in terms of left and right child.
- Therefore, the number of leaf nodes with exactly one child is 0.
Case 2: Internal Nodes
- Internal nodes are the nodes that have at least one child.
- In a binary tree, an internal node can have either one child (left or right child) or two children (both left and right child).
- However, in the given binary tree, every node has an odd number of descendants, which means every internal node must have two children.
- Therefore, there are no internal nodes with exactly one child.
Based on the above analysis, we can conclude that there are no nodes in the given binary tree that have exactly one child. Hence, the correct answer is option 'A' (0).
To solve this problem, let's consider some properties of binary trees and the given conditions:
1. Binary Tree: A binary tree is a tree data structure in which each node has at most two children, referred to as the left child and the right child.
2. Odd Number of Descendants: Every node in the given binary tree has an odd number of descendants, which means that the total number of nodes in the subtree rooted at any node is always odd.
3. Every Node is Its Own Descendant: The given condition states that every node is considered to be its own descendant. This means that we need to include each node itself while counting the number of descendants.
Now, let's analyze the possible cases for nodes with exactly one child:
Case 1: Leaf Nodes
- Leaf nodes are the nodes that do not have any children.
- Since every node is considered to be its own descendant, leaf nodes have exactly one child (itself).
- However, leaf nodes do not have any children in terms of left and right child.
- Therefore, the number of leaf nodes with exactly one child is 0.
Case 2: Internal Nodes
- Internal nodes are the nodes that have at least one child.
- In a binary tree, an internal node can have either one child (left or right child) or two children (both left and right child).
- However, in the given binary tree, every node has an odd number of descendants, which means every internal node must have two children.
- Therefore, there are no internal nodes with exactly one child.
Based on the above analysis, we can conclude that there are no nodes in the given binary tree that have exactly one child. Hence, the correct answer is option 'A' (0).
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Community Answer
In a binary tree with n nodes, every node has an odd number of descend...
It is mentioned that each node has odd number of descendants including node itself, so all nodes must have even number of descendants 0, 2, 4 so on. Which means each node should have either 0 or 2 children. So there will be no node with 1 child. Hence 0 is answer.
Following are few examples.
a
/ \
b c
/ \
b c
a
/ \
b c
/ \
d e
Such a binary tree is full binary tree (a binary tree where every node has 0 or 2 children).
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