CMPS 340 File Processing 
Examples of insertion/deletion in a B+-tree 

Consider the B+-tree $T$ illustrated below.  Let its B-tree ---which is 
comprised of the internal nodes--- have order four. 
Suppose that the leaves are large enough to hold up to four records.
In order to ensure that space utilization is no worse than about 50%,
we allow a minimum of two records in a leaf.
(Note that each record contains two fields, the first being the ordering
field (an integer) and the second being a character string.)

For each operation in the list (a) through (i),
show the B+-tree that results from performing that operation on T.

Assume, in carrying out the operations, that redistribution is used 
whenever possible.  That is, split an overflowing node only if all its
adjacent siblings are full.  Similarly, concatenate two nodes only if all
adjacent siblings of the underflowing node are on the verge of underflowing.
(Note: Redistribution is never to extend beyond two adjacent siblings!)

(a) insert 56 ink            (e) delete 48 
(b) insert 11 ink            (f) delete 50 
(c) insert 31 ink            (g) delete 61
(d) insert 20 ink            (h) delete 78 
                             (i) delete 96 


                          +----+----+----+
                          | 24   63   88 |
                          +----+----+----+
                         /     |     \    \
                    +---+      |      \    +-------------------+
                   /           |       +----------+             \
                  /            |                   \             \
    +----+---+----+        +----+----+----+         +----+       +----+
    | -2   6   19 |        | 34   50   58 |         | 80 |       | 96 |
    +----+---+----+        +----+----+----+         +----+       +----+
  /      |   |    |        |    |     \    \        |    |
 /       |   |    |        |    |      \    \        )  / 
+        |   |    |        |    |       +    \       | +
|  +-------+ | +---------+ | +--------+ | +--------+ | | +-------+   +--------+
|  | 1 bat | | | 19 cat  | | | 39 dog | | | 61 moo | | | | 88 is |   | 96 an  |
|  | 4 eel | | | 21 rat  | | | 44 cow | | | 62 due | | | | 91 of |   | 99 the |
|  +-------+ | | 22 clam | | | 48 ant | | +--------+ | | +-------+   +--------+
|            | | 23 kirk | | | 49 on  | |            | |
|            | +---------+ | +--------+ |            | +-----+
|            |             |            |            |        \
|            |             |            |            |         \
+--------+  +--------+   +--------+   +--------+   +--------+   +--------+
| -5 dam |  |  9 bob |   | 27 ted |   | 50 who |   | 75 arf |   | 81 boo | 
| -3 eat |  | 12 jim |   | 30 tom |   | 55 orc |   | 78 tim |   | 85 you |
+--------+  | 16 amy |   | 32 liz |   +--------|   +--------+   +--------+
            | 17 eek |   | 33 no  | 
            +--------+   +--------+