Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 151
root
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*
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Fig. 4.21. Tree of Fig. 4.21 represented as linked data structure
Before investigating how trees might be used advantageously and how to perform operations on trees,
we give an example of how a tree may be constructed by a program. Assume that a tree is to be generated
containing nodes with the values of the nodes being n numbers read from an input file. In order to make the
problem more challenging, let the task be the construction of a tree with n nodes and minimal height. In
order to obtain a minimal height for a given number of nodes, one has to allocate the maximum possible
number of nodes of all levels except the lowest one. This can clearly be achieved by distributing incoming
nodes equally to the left and right at each node. This implies that we structure the tree for given n as shown
in Fig. 4.22, for n = 1, ... , 7.
n=1 n=2 n=3
n=4 n=5
n=6
n=7
Fig. 4.22. Perfectly balanced trees
The rule of equal distribution under a known number n of nodes is best formulated recursively:
1. Use one node for the root.
2. Generate the left subtree with nl = n DIV 2 nodes in this way.
3. Generate the right subtree with nr = n - nl - 1 nodes in this way.
The rule is expressed as a recursive procedure which reads the input file and constructs the perfectly
balanced tree. We note the following definition: A tree is perfectly balanced, if for each node the numbers
of nodes in its left and right subtrees differ by at most 1.
TYPE Node = POINTER TO RECORD (* ADenS441_BalancedTree *)