$11\mathrm{.}3$ Construct a B$+-$tree for the following set of key values: $(2,3,5,7,11,17,19,23,29,31)$ Assume that the tree is initially empty and values are added in ascending order. Construct B$+-$trees for the cases where the number of pointers that will fit in one node is as follows: a$.$ Four b$.$ Six c$.$ Eight $11\mathrm{.}4$ For each B$+-$tree of Practice Exercise $11\mathrm{.}3,$ show the form of the tree after each of the following series of operations: a$.$ Insert $9.$ b$.$ Insert $10.$ c$.$ Insert $8.$ d$.$ Delete $23.$ e$.$ Delete $19.11\mathrm{.}5$ Consider the modified redistribution scheme for B$+-$trees described on page $501.$ What is the expected height of the tree as a function of n$?11\mathrm{.}6$ Suppose that we are using extendable hashing on a file that contains records with the following search$-$key values: $2,3,5,7,11,17,19,23,29,31$ Show the extendable hash structure for this file if the hash function is h$($x$)=$ x mod $8$ and buckets can hold three records. $11\mathrm{.}7$ Show how the extendable hash structure of Practice Exercise $11\mathrm{.}6$ changes as the result of each of the following steps: a$.$ Delete $11.$ b$.$ Delete $31.$ Practice Exercises $533$ c$.$ Insert $1.$ d$.$ Insert $15.$