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Algorithm 10.3 The Postorder Traversal of an Ordered Tree To traverse a nonempty ordered tree: 1. Do a recursive preorder traversal of each subtree in order. 2. Visit the root. Note that the level order and the preorder traversals always visit the root of each subtree first before visiting its other nodes. The postorder traversal always visits the root of each subtree last after visiting all of

its other nodes. Also, the preorder traversal always visits the right-most node last, while the postorder traversal always visits the left-most node first. The preorder and postorder traversals are recursive. They also can be implemented iteratively using a stack. The level order traversal is implemented iteratively using a queue. Review Questions

An increase in the labor supply, ceteris paribus, expands potential output. The law of diminishing returns shows that the incremental output from an additional labor input decreases when other economic resources and technology are unchanged. Thus, the possibility exists that aggregate output could increase while output per capita decreases. Expecting rapid population growth, economists in the early nineteenth century predicted such growth would result in declining output per capita. Thomas Malthus, in particular, held that the population would increase at such a rapid rate that the economy would increasingly be unable to grow enough food to feed its population; eventually output per capita would fall to a subsistence level. While technology has allowed highly industrialized countries to avoid these gloomy projections, rapid population growth is a problem for many developing countries.

101 All the classes in Java form a single tree, called the Java inheritance tree a What is the size of the Java inheritance tree in Java 13 b What is the root of the tree c What kind of node is a final class in the Java inheritance tree True or false: a The depth of a node in a tree is equal to the number of its ancestors b The size of a subtree is equal to the number of descendants of the root of the subtree c If x is a descendant of y, then the depth of x is greater than the depth of y d If the depth of x is greater than the depth of y, then x is a descendant of y e A tree is a singleton if and only if its root is a leaf f.

Every leaf of a subtree is also a leaf of its supertree g The root of a subtree is also the root of its supertree h The number of ancestors of a node equals its depth i If R is a subtree of S and S is a subtree of T, then R is a subtree of T j A node is a leaf if and only if it has degree 0 k In any tree, the number of internal nodes must be less than the number of leaf nodes l A tree is full if and only if all of its leaves are at the same level m Every subtree of a full binary tree is full n Every subtree of a complete binary tree is complete For the tree shown in Figure 1013, find: a all ancestors of node F b all descendants of node F c.

all nodes in the subtree rooted at F d all leaf nodes For each of the five trees shown in Figure 1014 on page 195, list the leaf nodes, the children of node C, the depth of node F, all the nodes at level 3, the height, and the order of the tree How many nodes are in the full tree of: a order 3 and height 4 b order 4 and height 3 c order 10 and height 4 d order 4 and height 10 Give the order of visitation of the tree shown in Example 102 on page 187 using the: a level order traversal b preorder traversal c postorder traversal.

CHAP. 10]

Note!

Which traversals always visit: a. the root first b. the left-most node first c. the root last d. the right-most node last The level order traversal follows the pattern as reading a page of English text: left-to-right, row-by-row. Which traversal algorithm follows the pattern of reading vertical columns from left to right Which traversal algorithm is used in the call tree for the solution to Problem 9.32 on page 184