Assume that you have just built a B+-tree index using Alternative (2) on a Heap file containing 40,000 records. That is, the actual records are stored in another file outside the index that has no particular order since it is a Heap file. Also, Alternative (2) means that we have one data entry at the leaf nodes of the B+-tree for every record. The search key field for this B+-tree index is a 40-byte string, and it is a candidate key (therefore unique). Pointers (i.e., record ids and page ids) are 10-byte values. The size of one disk page is 1000 bytes. The index T was built in a bottom-up fashion using the bulk-loading algorithm, and the nodes at each level were filled up as much as possible.

1. How many levels does the resulting tree T have?

2. For each level of the tree, how many nodes are at that level?

3. How many levels would the resulting tree T have if key compression is used and it reduces the average size of each key in an entry to 10 bytes?

4. How many levels would the resulting tree T have without key compression but with all pages 70 percent full?

5. Assume that each record is 100 bytes and is stored in the Heap file with disk pages of 1000 bytes each. Assume that you use the first B+-tree T that you computed, where the nodes were filled up as much as possible. Consider now a range query that uses this tree and has selectivity (reduction factor) 1% (i.e., 0.01). How many I/Os you need to perform in order to retrieve all the records that satisfy the range query using this tree? Explain.