Solution

The correct answer is 2.

Key Points

• A B⁺ tree is a type of self-balancing tree data structure that maintains sorted data and allows for efficient insertion, deletion, and search operations.
• In a B⁺ tree, nodes have a minimum and maximum number of keys, based on the order of the tree.
• For a B⁺ tree of order 5, the maximum number of keys in a node is 5.
• The minimum number of keys in a non-root node is calculated as ⌈m/2⌉, where m is the order of the tree.
• For m = 5, ⌈5/2⌉ = 2, hence the minimum number of keys in a non-root node is 2.

Step-by-Step Solution / Explanation

• Formula Used:
  • Minimum number of keys in a non-root node = ⌈m/2⌉
• Step-by-step substitution:
  • The order of the B⁺ tree is given as m = 5.
  • Substitute m = 5 into the formula: ⌈m/2⌉ = ⌈5/2⌉.
  • Perform the division: 5 ÷ 2 = 2.5.
  • Apply the ceiling function: ⌈2.5⌉ = 2.
• Intermediate calculations:
  • Order of the tree (m): 5.
  • Division result: 2.5.
  • Ceiling function applied: ⌈2.5⌉ = 2.
• Final computed result:
  • The minimum number of keys in a non-root node is 2.

Additional Information

• Properties of B⁺ Trees:
  • All leaf nodes are at the same level, ensuring balanced height.
  • Internal nodes act as a guide for searching keys but do not store actual data.
  • Leaf nodes contain pointers to the next leaf node, forming a linked list for sequential access.
• Applications of B⁺ Trees:
  • Used in databases and file systems for indexing and efficient data retrieval.
  • Supports range queries and sequential access efficiently.
  • Ideal for scenarios requiring balanced search trees with high fan-out.
• Advantages of B⁺ Trees:
  • Efficient searching, insertion, and deletion due to balanced structure.
  • Allows for fast sequential access to data via linked leaf nodes.
  • Minimizes disk I/O operations, making them suitable for large datasets.