Problem 1: `` Baron Harkonnen's Big Bad B-Tree Family" (30 pts)

(6) 1.1

Define a $B+$-tree of order m. That is, what conditions must all nodes in a $B+$-tree satisfy. Be sure to address both keys and records.

(6) 1.2

Define overflow in the context of a $B^+$-tree of order $m$, and explain briefly how the condition is corrected. Drawing a picture may be helpful here in explaining your answer.

(6) 1.3

Define underflow in the context of a $B^+$-tree of order $m$, and explain briefly how the condition is corrected. Drawing a picture may be helpful here in explaining your answer.

(6) 1.4

In performing a ``search-and-insert-if-record-not-found'' operation on a $B^+$-tree, the new key matched a value in the root node, yet the record was inserted.

Is the above scenario valid? If so, explain how. If not, explain why not.

(6) 1.5

Database applications typically use $B+$-trees where m is between 80 and 100. Give two reasons why. Explain your answer for full credit.