Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 153
prints the resulting tree: The empty tree results in no printing, the subtree at level L in first printing its own
left subtree, then the node, properly indented by preceding it with L tabs, and finally in printing its right
subtree.
4.4.2 Basic Operations on Binary Trees
There are many tasks that may have to be perfomed on a tree structure; a common one is that of
executing a given operation P on each element of the tree. P is then understood to be a parameter of the
more general task of visting all nodes or, as it is usually called, of tree traversal. If we consider the task as a
single sequential process, then the individual nodes are visited in some specific order and may be
considered as being laid out in a linear arrangement. In fact, the description of many algorithms is
considerably facilitated if we can talk about processing the next element in the tree based in an underlying
order. There are three principal orderings that emerge naturally from the structure of trees. Like the tree
structure itself, they are conveniently expressed in recursive terms. Referring to the binary tree in Fig. 4.24
in which R denotes the root and A and B denote the left and right subtrees, the three orderings are
1. Preorder: R, A, B (visit root before the subtrees)
2. Inorder: A, R, B
3. Postorder: A, B, R (visit root after the subtrees)
R
A
B
Fig. 4.24. Binary tree
Traversing the tree of Fig. 4.20 and recording the characters seen at the nodes in the sequence of
encounter, we obtain the following orderings:
1. Preorder: * + a / b c - d * e f
2. Inorder: a + b / c * d - e * f
3. Postorder: a b c / + d e f * - *
We recognize the three forms of expressions: preorder traversal of the expression tree yields prefix
notation; postorder traversal generates postfix notation; and inorder traversal yields conventional infix
notation, although without the parentheses necessary to clarify operator precedences.
Let us now formulate the three methods of traversal by three concrete programs with the explicit
parameter t denoting the tree to be operated upon and with the implicit parameter P denoting the operation
to be performed on each node. Assume the following definitions:
TYPE Node = POINTER TO RECORD ... left, right: Node END
The three methods are now readily formulated as recursive procedures; they demonstrate again the fact
that operations on recursively defined data structures are most conveniently defined as recursive algorithms.