Question $2.(30$ marks$)$

Consider the original $B+$ tree with order $d=4$ and height $h=3$ as follows:

When answering the following questions, please follow the assumptions:

A left pointer in an internal node guides towards keys smaller than its corresponding key, while a right pointer guided towards keys larger or equal than its corresponding key.

A leaf or internal node underflows when the number of pointers goes below $|$~$\frac{d}{2}$~$|.$

$(1)$ Insert $4$ into the original $B+$ tree. Draw the resulting tree. $[15$ marks$]$

$(2)$Delete $43$ from the original B$+$ tree. Draw the resulting tree. $[15$ marks$]$