Crude Method: create two versions of the B-tree insertion method. The first one is optimistic. It is the same as the method as shown below:

Except that lines 13-16 are deleted, and a new line is added after line 2 that checks if x is full, and if so, it does not do the insertion. Instead, it returns an error code. If the first method fails, the second method does the insertion exactly as stated in the book. Give an example that shows that this crude method can do unnecessary splits. Also state the time complexity of the algorithm.

B-TREE-INSERT-NONFULL (x, k) 2 if x. leaf while ?-1 and k ?x.keyi 4 x.key+1 = x.keyi x.keyi1 DISK-WRITE(x) 9 else while 1-1 and k ?x.key, 10 DISK-READ(x.ci) 12 13 14 15 16 17 B-TREE-SPLIT-CHILD (x, i) ifk?x.keyi B-TREE-INSERT-NONFULL(x??i, k) B-TREE-INSERT-NONFULL (x, k) 2 if x. leaf while ?-1 and k ?x.keyi 4 x.key+1 = x.keyi x.keyi1 DISK-WRITE(x) 9 else while 1-1 and k ?x.key, 10 DISK-READ(x.ci) 12 13 14 15 16 17 B-TREE-SPLIT-CHILD (x, i) ifk?x.keyi B-TREE-INSERT-NONFULL(x??i, k)