Variable Length Tree Structures Having Minimum Average Search Time
Sussenguth suggests in a paper (1963) that a
file should be organized as a doubly-chained tree 
structure if it is necessary both to search and to update
frequently.  Such a structure provides a compromise 
between the fast search/slow update characteristics of
binary searching and the slow search/fast update 
characteristics of serial searching.  His method, however,
contains the limiting restriction that all 
terminal nodes lie on the same level of the tree.  This paper
considers the effect of relaxing this restriction. 
 First, trees which have the property that a priori the
filial set of each node is well defined are studied. 
 It is proved that coding the nodes within each filial
set with respect to the number of terminal nodes 
reachable from each is necessary and sufficient to guarantee
minimum average search time.  Then the more 
general case (that is, where the entire structure of
the tree is changeable) is treated.  A procedure 
is developed for constructing a tree with a minimum
average search time.  A simple closed expression 
for this minimum average search time is obtained as
a function of the number of terminal nodes.  The 
storage capacity required to implement the doubly-chained
tree structure on a digital computer is also 
determined.  Finally, the total cost of the structure,
using Sussenguth's cost criterion, is computed. 
 It is shown that significant improvements in both
the average search time and the total cost can be 
obtained by relaxing Sussenguth's restriction that all
terminal nodes lie on the same level of the tree.
CACM February, 1969
Patt, Y. N.
