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? 


Question 3Answer

a.

4

b.

3

c.

5

d.

6

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.

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? a. 4, b. 3, c. 5, d. 6

Understand how node splitting occurs in a B-tree of order 4 after 10 successive insertions. Explore B-tree properties and how splitting propagates. This problem is ideal for students in university-level computer science courses.

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