GATE DA 2024 Set 1 — Question 52
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Programming, Data Structures & Algorithms → Trees → Tree Properties
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Question
Let , and represent height, number of internal nodes, number of leaf nodes, and the total number of nodes respectively in a rooted binary tree.Which of the following statements is/are always TRUE?
Correct answer
(A) L ≤ I + 1; (B) H + 1 ≤ N ≤ 2^(H+1) - 1; (C) H ≤ I ≤ 2^H - 1
Solution
We analyze each statement for a rooted binary tree:(A)
In a binary tree, let be the number of nodes with 2 children. The number of leaf nodes is given by . The total number of internal nodes includes nodes with 1 child () and nodes with 2 children (), so . Substituting , we get , which implies . Since , we have . This statement is always true.(B)
The minimum number of nodes in a binary tree of height occurs in a path graph (skewed tree), where . The maximum number of nodes occurs in a complete binary tree (or perfect binary tree), where . Thus, is always true.(C)
Consider a skewed tree (path graph) of height . It has only 1 leaf (). Here, implies , which is false. Thus, this statement is not always true.Therefore, statements (A), (B), and (C) are correct.
In a binary tree, let be the number of nodes with 2 children. The number of leaf nodes is given by . The total number of internal nodes includes nodes with 1 child () and nodes with 2 children (), so . Substituting , we get , which implies . Since , we have . This statement is always true.(B)
The minimum number of nodes in a binary tree of height occurs in a path graph (skewed tree), where . The maximum number of nodes occurs in a complete binary tree (or perfect binary tree), where . Thus, is always true.(C)
- Lower bound: To achieve height , there must be a path of length from the root to a leaf. This path involves edges and nodes. The first nodes on this path (including the root) must have at least one child, making them internal nodes. Thus, .
- Upper bound: The maximum number of internal nodes occurs in a perfect binary tree, where and . Then . Thus, .
Consider a skewed tree (path graph) of height . It has only 1 leaf (). Here, implies , which is false. Thus, this statement is not always true.Therefore, statements (A), (B), and (C) are correct.
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