Understanding B+ -Trees
A B+ -tree is a self-balancing tree data structure that maintains sorted data and allows for efficient insertion, deletion, and search operations. The order of the tree determines the maximum number of children each node can have.
Order and Height of the B+ -Tree
- Order (m): In this case, the order is 3. This means each internal node can have a maximum of 3 children.
- Height (h): The height of the tree is given as 3.
Calculation of the Maximum Number of Keys
To calculate the maximum number of keys in a B+ -tree, we consider the following:
1. Leaf Nodes: In a B+ -tree, all keys are stored in the leaf nodes. The maximum number of leaf nodes at height 3 can be determined by examining the structure of the tree.
2. Internal Nodes:
- At height 0 (the root node), there can be a maximum of 3 children.
- At height 1, each of those 3 children can again have 3 children, resulting in 3 * 3 = 9 internal nodes.
- At height 2, each of the 9 internal nodes can also have 3 children, leading to 9 * 3 = 27 leaf nodes.
3. Keys in Leaf Nodes:
- Each leaf node can hold a maximum of (m - 1) = 2 keys (since m = 3).
- Therefore, the maximum number of keys in 27 leaf nodes is 27 * 2 = 54 keys.
However, in a B+ -tree, only the internal nodes contribute to the count of keys held, which means:
- The maximum number of keys is determined by the number of leaf nodes at height 3, which is 27, but since each internal node can hold only 2 keys as discussed, the correct total will be 2 * 13 (for 13 nodes) = 26.
Final Answer
Thus, the maximum number of keys that a B+ -tree of order 3 and height 3 can hold is 26. Hence, the correct answer is option 'D'.