GATE DA 2026 Set 1 — Question 41
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Database Management & Warehousing → File Organization & Indexing → B+ Trees
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Question
Consider a B+ Tree where the maximum number of key values in each leaf node is 2 and the maximum number of pointers in each non-leaf node is 3. Let the content of the B+ Tree be as shown in the figure.
Which of the following options denotes the key value(s) stored in the root node after inserting a key value 3 in the given B+ Tree?
Which of the following options denotes the key value(s) stored in the root node after inserting a key value 3 in the given B+ Tree?
Correct answer
(A) 5
Solution
1.Initial State:
- Root:
[5, 8](2 keys, 3 pointers). The root is full (max pointers = 3 max keys = 2). - Leaves:
[1, 5],[7, 8],[9, 12]. Max keys per leaf = 2. - The tree structure implies a convention where the left pointer covers values and the right pointer covers values (since 5 is in the left leaf of 5).
- 3 is , so it goes to the leftmost leaf
[1, 5]. - Leaf becomes
[1, 3, 5]. This exceeds the capacity (2). - Leaf Split: The leaf
[1, 3, 5]splits. Typically, B+ trees split at the median. Let's assume a split into[1, 3]and[5]. The separator key (smallest in the right node or appropriate separator) is inserted into the parent. Let's say 3 or 5 is promoted. Given the existing keys, let's assume standard behavior where the middle element (3) is pushed up as a separator, or the split is[1]and[3, 5]promoting 3. In either case, a key (say 3) is inserted into the parent.
- The parent (Root) currently has
[5, 8]. It is full. - We attempt to insert the promoted key (e.g., 3) into the root.
- Temporary Root keys:
[3, 5, 8](sorted). - Root Split: Since the root is full, it must split. The middle key of
[3, 5, 8]is 5. - The middle key 5 is promoted to be the new root.
- The old root splits into two internal nodes:
[3]and[8].
- The new root contains the single key 5.
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