In a B+ - tree where each node can hold at most four key values, a root to leaf path consists of the following nodes:
A = (49, 77, 83, –), B = (7, 19, 33, 44), C = (20*, 22*, 25*, 26*)
The *-marked keys signify that these are data entries in a leaf. Assume that a pointer between keys k1​ and k2​ points to a subtree containing keys in [k1,k2], and that when a leaf is created, the smallest key in it is copied up into A. A record with key value 23 is inserted into the B+ - tree. The smallest key value in the parent of the leaf that contains 25* is ______.

The correct answer is 33.

Step-by-Step Explanation:

1. Initial State of the B+ Tree:
We are given the following path from root to leaf:
- Root node A = (49, 77, 83, –)
- Internal node B = (7, 19, 33, 44)
- Leaf node C = (20*, 22*, 25*, 26*)
Note that leaf node C is currently full because it contains four key values, which is the maximum limit per node.

2. Inserting Key 23:
When a record with key value 23 is inserted, it must be placed in leaf node C because 20 < 23 < 26. This temporarily creates an overfull leaf node containing 5 keys: (20*, 22*, 23*, 25*, 26*).

3. Splitting Leaf Node C:
To resolve the overflow, leaf node C is split into two leaf nodes:
- Left leaf C₁ = (20*, 22*, 23*)
- Right leaf C₂ = (25*, 26*) (which contains 25*)
Following B+ tree insertion rules, the smallest key of the new right leaf C₂ (which is 25) is copied up to the parent internal node B.

4. Splitting Internal Node B:
With the entry 25 copied up, the parent node B now contains 5 keys: (7, 19, 25, 33, 44). Since this exceeds the maximum limit of 4 keys, internal node B must split:
- The middle key (25) is pushed up to the root node A.
- The remaining keys are split into two internal nodes:
  • Left internal node B₁ = (7, 19)
  • Right internal node B₂ = (33, 44)

5. Identifying the Parent of Leaf C₂:
The pointer corresponding to the range [25, 33] resides in the right split internal node B₂ = (33, 44). Therefore, the leaf node containing 25* (which is C₂) becomes a child of B₂.
The keys present in the parent node B₂ are (33, 44), making the smallest key value in this parent node 33.