A B-tree of order 4 is built from scratch by 10 successive insertions. What is the maximum number of node-splitting operations that may take place?

To answer this question, we need to understand how a B-tree works. A B-tree of order 4 means that each node in the tree can hold a maximum of 3 keys and 4 children.

1. Initially, the B-tree is empty. The first key is inserted without any splitting.

2. The second and third keys are also inserted without any splitting since the root node can hold up to 3 keys.

3. The fourth key causes the first split. The root node is split into two nodes, each holding one key, and a new root node is created holding the key that maintains the B-tree properties.

4. The fifth, sixth, and seventh keys are inserted without any splitting since the two child nodes can each hold up to 3 keys.

5. The eighth key causes the second split. One of the child nodes is split into two nodes, each holding one key, and the key that maintains the B-tree properties is moved up to the parent node.

6. The ninth key is inserted without any splitting since there is still room in the child nodes.

7. The tenth key causes the third split. One of the child nodes is split into two nodes, each holding one key, and the key that maintains the B-tree properties is moved up to the parent node.

So, the maximum number of node-splitting operations that may take place when building a B-tree of order 4 by 10 successive insertions is 3.