Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 187
END
END delete;
PROCEDURE ShowTree (VAR W: Texts.Writer; p: Page; level: INTEGER);
VAR i: INTEGER;
BEGIN
IF p # NIL THEN
FOR i := 1 TO level DO Texts.Write(W, 9X) END;
FOR i := 0 TO p.m-1 DO Texts.WriteInt(W, p.e[i].key, 4) END;
Texts.WriteLn(W);
IF p.m > 0 THEN ShowTree(W, p.p0, level+1) END;
FOR i := 0 TO p.m-1 DO ShowTree(W, p.e[i].p, level+1) END
END
END ShowTree;
Extensive analysis of B-tree performance has been undertaken and is reported in the referenced article
(Bayer and McCreight). In particular, it includes a treatment of the question of optimal page size, which
strongly depends on the characteristics of the storage and computing system available.
Variations of the B-tree scheme are discussed in Knuth, Vol. 3, pp. 476-479. The one notable
observation is that page splitting should be delayed in the same way that page merging is delayed, by first
attempting to balance neighboring pages. Apart from this, the suggested improvements seem to yield
marginal gains. A comprehensive survey of B-trees may be found in [4-8].
4.7.2 Binary B-Trees
The species of B-trees that seems to be least interesting is the first order B-tree (n = 1). But sometimes
it is worthwhile to pay attention to the exceptional case. It is plain, however, that first-order B-trees are not
useful in representing large, ordered, indexed data sets involving secondary stores; approximately 50% of
all pages will contain a single item only. Therefore, we shall forget secondary stores and again consider the
problem of search trees involving a one-level store only.
A binary B-tree (BB-tree) consists of nodes (pages) with either one or two items. Hence, a page
contains either two or three pointers to descendants; this suggested the term 2-3 tree. According to the
definition of B-trees, all leaf pages appear at the same level, and all non-leaf pages of BB-trees have either
two or three descendants (including the root). Since we now are dealing with primary store only, an
optimal economy of storage space is mandatory, and the representation of the items inside a node in the
form of an array appears unsuitable. An alternative is the dynamic, linked allocation; that is, inside each
node there exists a linked list of items of length 1 or 2. Since each node has at most three descendants and
thus needs to harbor only up to three pointers, one is tempted to combine the pointers for descendants and
pointers in the item list as shown in Fig. 4.47. The B-tree node thereby loses its actual identity, and the
items assume the role of nodes in a regular binary tree. It remains necessary, however, to distinguish
between pointers to descendants (vertical) and pointers to siblings on the same page (horizontal). Since
only the pointers to the right may be horizontal, a single bit is sufficient to record this distiction. We
therefore introduce the Boolean field h with the meaning horizontal. The definition of a tree node based on
this representation is given below. It was suggested and investigated by R. Bayer [4-3] in 1971 and
represents a search tree organization guaranteeing p = 2*log(N) as maximum path length.
TYPE Node = POINTER TO RECORD
key: INTEGER;
...........
left, right: Node;
h: BOOLEAN (*right branch horizontal*)