The maximum number of leaf nodes that a B+ tree of height 4 and order 3 can contain is 54. This is calculated by multiplying the maximum number of nodes at each level by the order of the tree, starting from the root.
The question pertains to a B+ tree, a type of self-balancing tree data structure that maintains sorted data in a way that allows for efficient insertion, deletion, and sequential access operations. One must calculate the maximum number of leaf nodes that a B+ tree can contain given a height of 4 and an order of 3. In a B+ tree of order 'm', each internal node can have a maximum of 'm' children. The leaf nodes do not store any keys, only the values or pointers to the data, and the keys are stored in internal nodes which point to leaf nodes that contain the actual records.
Therefore, excluding the root, each level can have up to 'm' times the number of nodes of the previous level. Starting with a root node that has a minimum of 2 children, each subsequent level multiplies the number of possible nodes by the tree order, up to the height of the tree. For a B+ tree of height 4 and order 3, this would be calculated as follows:
- Level 1 (root): At least 2 nodes (by definition, the root has at least 2 children)
- Level 2: Up to 2 * 3 = 6 nodes
- Level 3: Up to 6 * 3 = 18 nodes
- Level 4 (leaf level): Up to 18 * 3 = 54 leaf nodes
So, the B+ tree can contain a maximum of 54 leaf nodes.