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a, b, c, d, e, f, g, h, i, j, k, l, m.
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Figure 10.10 A level order traversal
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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.
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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, b, e, h, i, f, c, d, g, j, k, l, m.
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Figure 10.11 A preorder traversal
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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
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The postorder traversal of the tree shown in Figure 10.12 would visit the nodes in the following order:
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h, i, e, f, b, c, j, k, l, m, g, d, a.
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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
Figure 10.12 A tree
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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
10.1 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 1.3 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 10.13, 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 10.14 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 10.2 on page 187 using the: a. level order traversal b. preorder traversal c. postorder traversal
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