Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 182
As a consequence, the new root page contains a single item only. The details can be gathered from the
program presented below, and Fig. 4.45 shows the result of using the program to construct a B-tree with
the following insertion sequence of keys:
20; 40 10 30 15; 35 7 26 18 22; 5; 42 13 46 27 8 32; 38 24 45 25;
The semicolons designate the positions of the snapshots taken upon each page allocation. Insertion of the
last key causes two splits and the allocation of three new pages.
a)
20
b)
20
10 15 30 40
c)
20 30
7 10 15 18 22 26 35 40
d)
10 20 30
5 7 15 18 22 26 35 40
e)
10 20 30 40
5 7 8 13 15 18 22 26 27 32 35 42 46
f)
25
10 20
30 40
5 7 8 13 15 18 22 24 26 27 32 35 38 42 45 46
Fig. 4.45. Growth of B-tree of order 2
Since each activation of search implies one page transfer to main store, k = log n (N) recursive calls are
necessary at most, if the tree contains N items. Hence, we must be capable of accommodating k pages in
main store. This is one limiting factor on the page size 2n. In fact, we need to accommodate even more
than k pages, because insertion may cause page splitting to occur. A corollary is that the root page is best
allocated permanently in the primary store, because each query proceeds necessarily through the root
page.
Another positive quality of the B-tree organization is its suitability and economy in the case of purely
sequential updating of the entire data base. Every page is fetched into primary store exactly once.
Deletion of items from a B-tree is fairly straight-forward in principle, but it is complicated in the details.
We may distinguish two different circumstances:
1. The item to be deleted is on a leaf page; here its removal algorithm is plain and simple.
2. The item is not on a leaf page; it must be replaced by one of the two lexicographically adjacent
items, which happen to be on leaf pages and can easily be deleted.
In case 2 finding the adjacent key is analogous to finding the one used in binary tree deletion. We descend
along the rightmost pointers down to the leaf page P, replace the item to be deleted by the rightmost item
on P, and then reduce the size of P by 1. In any case, reduction of size must be followed by a check of the