A B-tree of order 4 means that each node can have a maximum of 4 children and, hence, up to 3 keys. When building a B-tree from scratch and inserting keys successively, splitting occurs when a node becomes full (contains 3 keys) and a new key needs to be inserted.

Let’s work through the problem to determine the maximum number of splits that could occur.

Step-by-Step Analysis:

1. First Node: Initially, the B-tree is empty. The first 3 insertions will fill the root node without any splits.
2. Splitting the Root: When the 4th key is inserted, the root becomes full, and we split it, creating two child nodes. This is the first split.
3. Subsequent Insertions: After splitting the root, further insertions can fill up the child nodes. Once a child node reaches 3 keys, it will need to split as well.
4. Cascading Splits: When a child node splits, if the parent node is also full, this can cause a split in the parent as well. Therefore, splits can propagate upwards to higher levels.

Maximum Scenario:

• The root will split once, and then there could be cascading splits if the child nodes keep filling up.
• In the worst-case scenario, every time a split happens, the number of nodes increases, but no unnecessary rearrangements occur that would avoid further splits.
• To find the maximum number of splits, we can observe that at most logarithmic depth of splits occurs as the tree grows, particularly in the first few levels.

Conclusion:

The maximum number of node splits in this case, considering the B-tree of order 4 and 10 insertions, is 3.

Thus, the correct answer is:

b. 3

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Would you like any further explanation on how B-trees handle splitting or how this problem works? Here are some related questions:

1. What is the structure of a B-tree of order 4?
2. How does the process of node splitting work in a B-tree?
3. Why does splitting propagate upwards in a B-tree?
4. How does the insertion process differ in B-trees of higher or lower orders?
5. What are the advantages of using a B-tree for database indexing?

Tip: In B-trees, keeping track of key counts and how they propagate during insertions is essential for understanding their efficiency in storing large data sets.