Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 190
B
(LL)
B C A B C
A
C
A
B
(LR)
A C A B C
A
C
B
B
(RR)
A B A B C
A
C
C
B
(RL)
A C A B C
A
C
B
Fig. 4.49. Insertion in SBB-trees
It is advisable to stick no longer to the notion of pages out of which this organization had developed, for
we are only interested in bounding the maximum path length to 2*log(N). For this we need only ensure that
two horizontal pointers may never occur in succession on any search path. However, there is no reason to
forbid nodes with horizontal pointers to the left and right, i.e. to treat the left and right sides differently. We
therefore define the symmetric binary B-tree as a tree that has the following properties:
1. Every node contains one key and at most two (pointers to) subtrees.
2. Every pointer is either horizontal or vertical. There are no two consecutive horizontal pointers on any
search path.
3. All terminal nodes (nodes without descendants) appear at the same (terminal) level.
From this definition it follows that the longest search path is no longer than twice the height of the tree.
Since no SBB-tree with N nodes can have a height larger than log(N), it follows immediately that 2*log(N)
is an upper bound on the search path length. In order to visualize how these trees grow, we refer to Fig.
4.50. The lines represent snapshots taken during the insertion of the following sequences of keys, where
every semicolon marks a snapshot.