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for (it = tree.new InOrder(); it.hasNext(); ) { System.out.print(it.next() + " "); } System.out.print("\nPostOrder Traversal: "); for (it = tree.new PostOrder(); it.hasNext(); ) { System.out.print(it.next() + " "); } System.out.print("\nLevelOrder Traversal: "); for (it = tree.new LevelOrder(); it.hasNext(); ) { System.out.print(it.next() + " "); } System.out.println(); } }
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tree = [A, B, D, G, E, C, PreOrder Traversal: A B InOrder Traversal: D G PostOrder Traversal: G D LevelOrder Traversal: A B F, H, D G E B E A E B H C D E I] C F H F I F F G H I C H I C A I
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Each of the four iterators traverses the tree according to the algorithm that it implements.
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FORESTS A forest is a sequence of disjoint ordered trees. EXAMPLE 11.25 A Forest
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Figure 11.23 shows a forest that consists of three trees.
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Figure 11.23 A forest
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The following algorithm shows how a forest can be represented by a single binary tree. Algorithm 11.8 The Natural Mapping of a Forest into a Binary Tree 1. Map the root of the first tree into the root of the binary tree. 2. If node X maps into X' and node Y is the first child of X, then map Y into the left child of X'. 3. If node X maps into X' and node Z is the sibling of X, then map Z into the right child of X'. The roots of the trees themselves are considered siblings.
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BINARY TREES
[CHAP. 11
EXAMPLE 11.26 Mapping a Forest into a Binary Tree
Figure 11.24 is the mapping of the forest shown in Example 11.25. For example, in the original forest, C has oldest child F and next sibling D. In the corresponding binary tree, C has left child F and right child D.
Figure 11.24 The natural mapping of a forest into a binary tree
Review Questions
11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 How many leaf nodes does the full binary tree of height h = 3 have How many internal nodes does the full binary tree of height h = 3 have How many nodes does the full binary tree of height h = 3 have How many leaf nodes does a full binary tree of height h = 9 have How many internal nodes does a full binary tree of height h = 9 have How many nodes does a full binary tree of height h = 9 have What is the range of possible heights of a binary tree with n = 100 nodes Why is there no inorder traversal for general trees True or false: a. If all of its leaves are at the same level, then the binary tree is full. b. If the binary tree has n nodes and height h, then h lg n . c. A binary tree cannot have more than 2d nodes at depth d. d. If every proper subtree of a binary tree is full, then the tree itself must also be full.
Problems
11.1 11.2 For each of the binary trees in Figure 11.25 on page 221, draw the equivalent version that satisfies the second definition, namely that every internal node has two children. Give the order of visitation of the binary tree shown in Figure 11.26 using the specified traversal algorithm: a. the level order traversal b. the preorder traversal c. the inorder traversal d. the postorder traversal
CHAP. 11]
BINARY TREES
Figure 11.25 Binary trees
Figure 11.26 A binary tree
Figure 11.27 A binary tree
Give the order of visitation of the binary tree of size 10 shown in Example 11.1 on page 201 using: a. the level order traversal b. the preorder traversal c. the inorder traversal d. the postorder traversal Give the order of visitation of the binary tree shown in Figure 11.27 using: a. the level order traversal b. the preorder traversal c. the inorder traversal d. the postorder traversal
BINARY TREES
[CHAP. 11
11.5 11.6 11.7 11.8
Show the array that is obtained by using the natural mapping to store the binary tree shown in Problem 11.1. Show the array that is obtained by using the natural mapping to store the binary tree shown in Example 11.1 on page 201. Show the array that is obtained by using the natural mapping to store the binary tree shown in Problem 11.4. If the nodes of a binary tree are numbered according to their natural mapping, and the visit operation prints the node s number, which traversal algorithm will print the numbers in order Draw the expression tree for a*(b + c)*(d*e + f).
11.10 Write the prefix and the postfix representations for the expression in Problem 11.8. 11.11 Draw the expression tree for each of the prefix expressions given in Problem 5.2 on page 111. 11.12 Draw the expression tree for each of the infix expressions given in Problem 5.4 on page 111. 11.13 Draw the expression tree for each of the postfix expressions given in Problem 5.6 on page 111. 11.14 Draw the expression tree for the expression a*(b + c)*(d*e + f). 11.15 What are the bounds on the number n of nodes in a binary tree of height 4 11.16 What are the bounds on the height h of a binary tree with 7 nodes 11.17 What form does the highest binary tree have for a given number of nodes 11.18 What form does the lowest binary tree (i.e., the least height) have for a given number of nodes 11.19 Verify the recursive definition of binary trees (page 200) for the binary tree shown in Figure 11.28. 11.20 Draw all 42 binary trees of size n = 5. 11.21 How many different binary trees of size n = 6 are there 11.22 Derive a recurrence relation for the number f(n) of binary trees of size n.
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