Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 150
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Fig. 4.19. Ternary tree extended with special nodes
Of particular importance are the ordered trees of degree 2. They are called binary trees. We define an
ordered binary tree as a finite set of elements (nodes) which either is empty or consists of a root (node)
with two disjoint binary trees called the left and the right subtree of the root. In the following sections we
shall exclusively deal with binary trees, and we therefore shall use the word tree to mean ordered binary
tree. Trees with degree greater than 2 are called multiway trees and are discussed in sect. 4.7.1.
Familiar examples of binary trees are the family tree (pedigree) with a person's father and mother as
descendants (!), the history of a tennis tournament with each game being a node denoted by its winner and
the two previous games of the combatants as its descendants, or an arithmetic expression with dyadic
operators, with each operator denoting a branch node with its operands as subtrees (see Fig. 4.20).
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Fig. 4.20. Tree representation of expression (a + b/c) * (d – e*f)
We now turn to the problem of representation of trees. It is plain that the illustration of such recursive
structures in terms of branching structures immediately suggests the use of our pointer facility. There is
evidently no use in declaring variables with a fixed tree structure; instead, we define the nodes as variables
with a fixed structure, i.e., of a fixed type, in which the degree of the tree determines the number of pointer
components referring to the node's subtrees. Evidently, the reference to the empty tree is denoted by NIL.
Hence, the tree of Fig. 4.20 consists of components of a type defined as follows and may then be
constructed as shown in Fig. 4.21.
TYPE Node = POINTER TO NodeDesc;
TYPE NodeDesc = RECORD op: CHAR; left, right: Node END