Q$3.$ Consider a B$+$ tree shown in Figure below. We next wish to insert a tuple r $11$ with the key value $112.$ In which page will the insertion be performed; will that insertion result the split of the leaf node on which the record is inserted? Will the root be modified$/$split$?$ Which keys will be in the modified$/$new root?

a$)$ The record will be inserted on leaf page N$5.$ The insertion will not cause the split of N$5.$ The root note N$1$ will remain the same as shown in the figure.

b$)$ The record will be inserted on leaf page N$5.$ The insertion will not cause the split of N$5.$ The root node will not be split, but it will be modified by changing the key value $121$ to $107$ so as to correctly route the queries to the right node. Thus, the new root will have the keys

c$)$ The record will be inserted on node N$4.$ To create space for the new record, the key value $76($and corresponding record pointer$)$ will be shifted from node N$4$ to N$3.$ This will not result in a split of either N$4$ or root. The root, however, will be modified by replacing $76$ by $99.$ Thus, the keys in the root will be

d$)$ The record will be inserted on node N$4.$ Node N$4$ will split which will subsequently cause the existing root node to split increasing the height of the tree. The new root will contain the key value $106.$

Q$3.$ Consider a B$+$ tree shown in Figure below. We next wish to insert a tuple r $11$ with the key value $112.$ In which page will the insertion be performed; will that insertion result the split of the leaf node on which the record is inserted? Will the root be modified$/$split$?$ Which keys will be in the modified$/$new root?

a$)$ The record will be inserted on leaf page N$5.$ The insertion will not cause the split of N$5.$ The root note N$1$ will remain the same as shown in the figure.

b$)$ The record will be inserted on leaf page N$5.$ The insertion will not cause the split of N$5.$ The root node will not be split, but it will be modified by changing the key value $121$ to $107$ so as to correctly route the queries to the right node. Thus, the new root will have the keys

c$)$ The record will be inserted on node N$4.$ To create space for the new record, the key value $76($and corresponding record pointer$)$ will be shifted from node N$4$ to N$3.$ This will not result in a split of either N$4$ or root. The root, however, will be modified by replacing $76$ by $99.$ Thus, the keys in the root will be

d$)$ The record will be inserted on node N$4.$ Node N$4$ will split which will subsequently cause the existing root node to split increasing the height of the tree. The new root will contain the key value $106.$