Five node splitting operations occur when an entry is inserted into a B-tree. Then how many nodes are written?

To determine how many nodes are written when five node splitting operations occur in a B-tree during an entry insertion, we need to analyze the B-tree structure and the consequences of splitting nodes.

1. Break Down the Problem

• Node Split: When a node in a B-tree is full and a new entry is inserted, the node is split into two nodes. The median value is pushed up to the parent node.
• Insertion Process: Insertion in a B-tree may cause multiple nodes to become full, necessitating splits and upward propagation of keys.

2. Relevant Concepts

• For each node that splits, at least:
  • One node will be written (the original node that splits).
  • One new node is created and written (the newly formed node after the split).
  • The parent node is affected by the split and needs to be updated and written as well.

3. Analysis and Detail

1. First Split:

   • Write original node → 1 write
   • Write new node → 1 write
   • Write parent node → 1 write (if it had space)

   Total for first split = 3 writes.

2. Subsequent Splits: Continuing this process for each of the five splits will similarly result in multiple writes where:

   • The node that splits contributes at least 1 write.
   • The newly created node contributes 1 write.
   • The parent node could contribute 1 write, but if it needs to split too, additional writes would follow.

Assuming all splits (5 splits) cause parent writes, the calculation will follow through.

• If we denote the number of splits as $s$: $\text{Total writes} = s \times 3 \text{ (considering parent writes)}$

Thus, for five splits, we have: $\text{Total writes} = 5 \times 3 = 15$

4. Verify and Summarize

Calculating through each split and assuming that each split results in the corresponding writes provides a consistent total. This varies if we consider whether parent nodes can absorb additional splits without writing or if there are fewer levels in the tree.

Final Answer

Therefore, when five node splitting operations occur during an entry insertion in a B-tree, a total of 15 nodes are written.