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The ordered tree on the left has root node a and subtree sequence ( (b, ), (c, (d, ) ) ). The ordered tree on the right has root node a and subtree sequence ( (c, (d, ) ), (b, ) ). These two subtree sequences have the same elements, but not in the same order. Thus the two ordered trees are not the same.

Strict adherence to the definition reveals a subtlety often missed, as illustrated by the next example. EXAMPLE 10.6 Unequal Ordered Trees

Economic growth is concerned with the expansion of an economy s ability to produce (potential GDP) over time. Expansion of potential output occurs when there is an increase in natural resources, human resources, or capital, or when there is a technological advance. The two most common measures of economic growth are an increase in real GDP and an increase in output per capita. Of these two measures, an increase in output per capita is more meaningful since it indicates there are more goods and services available per person and hence a rise in the economy s standard of living. An increase in potential

The two trees T1 = (a, (B, C)) and T2 = (a, (B, , C)) are not the same ordered trees, even though they would probably both be drawn the same, as shown in Figure 10.9.

All the terminology for unordered trees applies the same way to ordered trees. In addition, we can also refer to the first child and the last child of a node in an ordered tree. It is sometimes useful to think analogously of a human genealogical tree, where the children are ordered by age: oldest first and youngest last. TRAVERSAL ALGORITHMS A traversal algorithm is a method for processing a data structure that applies a given operation to each element of the structure. For example, if the operation is to print the contents of the element, then the traversal would print every element in the structure. The process of applying the operation to an element is called visiting the element. So executing the traversal algorithm causes each element in the structure to be visited. The order in which the elements are visited depends upon which traversal algorithm is used. There are three common algorithms for traversing a general tree. The level order traversal algorithm visits the root, then visits each element on the first level, then visits each element on the second level, and so forth, each time visiting all the elements on one level before going down to the next level. If the tree is drawn in the usual manner with its root at the top and leaves near the bottom, then the level order pattern is the same left-to-right top-to-bottom pattern that you follow to read English text. EXAMPLE 10.7 The Level Order Traversal

The level order traversal of the tree shown in Figure 10.10 would visit the nodes in the following order:

13:

Algorithm 10.1 The Level Order Traversal of an Ordered Tree To traverse a nonempty ordered tree: 1. Initialize a queue. 2. Enqueue the root. 3. Repeat steps 4 7 until the queue is empty. 4. Dequeue node x from the queue. 5. Visit x. 6. Enqueue all the children of x in order.

CHAP. 10]

output can be conceptualized by an outward shift of an economy s production-possibility frontier. In our discussion of economic growth, we assume that increases in potential output are matched by equal increases in aggregate spending so that full-employment growth is assured.

The preorder traversal algorithm visits the root first and then does a preorder traversal recursively to each subtree in order. EXAMPLE 10.8 The Preorder Traversal The preorder traversal of the tree shown in Figure 10.11 would visit the nodes in this order: a,

Note that the preorder traversal of a tree can be obtained by circumnavigating the tree, beginning at the root and visiting each node the first time it is encountered on the left. Algorithm 10.2 The Preorder Traversal of an Ordered Tree To traverse a nonempty ordered tree: 1. Visit the root. 2. Do a recursive preorder traversal of each subtree in order. The postorder traversal algorithm does a postorder traversal recursively to each subtree before visiting the root. EXAMPLE 10.9 The Postorder Traversal

The postorder traversal of the tree shown in Figure 10.12 would visit the nodes in the following order: