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A stochastic process is a process that can be analyzed by a transition diagram, that is, it can be decomposed into sequences of events whose conditional probabilities can be computed. The game of craps is actually an infinite stochastic process since there is no limit to the number of events that could occur. As with the analysis in Example 10.4, most infinite stochastic processes can be reformulated into an equivalent finite stochastic process that is amenable to (finite) computers. Note that, unlike other tree models, decision trees and transition trees are usually drawn from left to right to suggest the time-dependent movement from one node to the next. ORDERED TREES Here is the recursive definition of an ordered tree: An ordered tree is either the empty set or a pair T = (r, S), where r is a node and S is a sequence of disjoint ordered trees, none of which contains r. The node r is called the root of the tree T, and the elements of the sequence S are its subtrees. The sequence S of course may be empty, in which case T is a singleton. The restriction that none of the subtrees contains the root applies recursively: x cannot be in any subtree, or in any subtree of any subtree, and so on. Note that this definition is the same as that for unordered trees except for the facts that the subtrees are in a sequence instead of a set and an ordered tree may be empty. Consequently, if two unordered trees have the same subsets, then they are equal; but as ordered trees, they won t be equal unless their equal subtrees are in the same order. Also subtrees of an ordered set may be empty. EXAMPLE 10.5 Unequal Ordered Trees
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The two trees shown in Figure 10.8 are not equal as ordered trees.
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Figure 10.8 Unequal ordered trees
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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.
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Strict adherence to the definition reveals a subtlety often missed, as illustrated by the next example. EXAMPLE 10.6 Unequal Ordered Trees
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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.
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Figure 10.9 A tree
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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: