$4\mathrm{.}41$ a$.$ Write a routine to perform insertion into a B$-$tree.

b$.$ Write a routine to perform deletion from a B$-$tree. When an item is deleted, is it

necessary to update information in the internal nodes?

c$.$ Modify your insertion routine so that if an attempt is made to add into a node

that already has M entries, a search is performed for a sibling with less than M

children before the node is split. $($use c$++)$