PLEASE JUST DO PROBLEM $3$: Problem $1(20$ points$)$

Construct a B$+$ tree for the following set of key values:

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$(2,3,5,7,11,17,19,23,29,31)$

$\backslash ]$

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 $4.\backslash ((\backslash$mathrm$\{$n$\}=4)\backslash )$

Note: Please illustrate the final structure of the B$+$ tree, showing the key distribution in each node after all insertions.

Problem $2(60$ points$)$

For the tree in Problem $1,$ show the form of the tree after each of the following series of operations:

$-$ Insert $9$

$-$ Insert $10$

$-$ Insert $8$

$-$ Delete $23$

$-$ Delete $19$

Note: Draw the updated B$+$ tree after each operation, showing changes in node structures and any necessary rebalancing.

Problem $3(20$ points$)$

What is the minimum space utilization for a $\backslash (\backslash$mathrm$\{$B$\}+\backslash )$ tree index?