vb.net print barcode zebra Figure 11.12 A binary tree in Java

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Figure 11.12 A binary tree
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Figure 11.13 The natural mapping of an incomplete binary tree
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BINARY TREE TRAVERSAL ALGORITHMS The three traversal algorithms that are used for general trees (see 10) apply to binary trees as well: the preorder traversal, the postorder traversal, and the level order traversal. In addition, binary trees support a fourth traversal algorithm: the inorder traversal. These four traversal algorithms are given next. Algorithm 11.2 The Level Order Traversal of a Binary Tree To traverse a nonempty binary tree: 1. Initialize a queue. 2. Enqueue the root. 3. Repeat steps 4 7 until the queue is empty. 4. Dequeue a node x from the queue. 5. Visit x. 6. Enqueue the left child of x if it exists. 7. Enqueue the right child of x if it exists. EXAMPLE 11.13 The Level Order Traversal of a Binary Tree
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Figure 11.14 on page 207 shows how the level order traversal looks on the full binary tree of height 3.
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Figure 11.14 The level order traversal of a binary tree
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The nodes are visited in the order A, B, C, D, E, F, G, H, I, J, K, L, M, N, O.
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Algorithm 11.3 The Preorder Traversal of a Binary Tree To traverse a nonempty binary tree: 1. Visit the root. 2. If the left subtree is nonempty, do a preorder traversal on it. 3. If the right subtree is nonempty, do a preorder traversal on it. EXAMPLE 11.14 The Preorder Traversal of a Binary Tree
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Figure 11.15 on page 208 shows the preorder traversal on the full binary tree of height 3.
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Figure 11.15 The preorder traversal of a binary tree
The nodes are visited in the order A, B, D, H, I, E, J, K, C, F, L, M, G, N, O.
Figure 11.16 shows how the preorder traversal of a binary 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:
Figure 11.16 The preorder traversal of a binary tree
Algorithm 11.4 The Postorder Traversal of a Binary Tree To traverse a nonempty binary tree: 1. If the left subtree is nonempty, do a postorder traversal on it. 2. If the right subtree is nonempty, do a postorder traversal on it. 3. Visit the root. EXAMPLE 11.15 The Postorder Traversal of a Binary Tree
Figure 11.17 shows the postorder traversal looks on the full binary tree of height 3.
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BINARY TREES
Figure 11.17 The postorder traversal of a binary tree
The nodes are visited in the order H, I, D, J, K, E, B, L, M, F, N, O, G, C, A.
The preorder traversal visits the root first and the postorder traversal visits the root last. This suggests a third alternative for binary trees: Visit the root in between the traversals of the two subtrees. That is called the inorder traversal. Algorithm 11.5 The Inorder Traversal of a Binary Tree To traverse a nonempty binary tree: 1. If the left subtree is nonempty, do a preorder traversal on it. 2. Visit the root. 3. If the right subtree is nonempty, do a preorder traversal on it. EXAMPLE 11.16 The Inorder Traversal of a Binary Tree
Figure 11.18 shows how the inorder traversal looks on the full binary tree of height 3.
Figure 11.18 The inorder traversal of a binary tree
The nodes are visited in the order H, D, I, B, J, E, K, A, L, F, M, C, N, G, O.
EXPRESSION TREES An arithmetic expression such as (5 x)*y + 6/(x + z) is a combination of arithmetic operators (+, , *, /, etc.), operands (5, x, y, 6, z, etc.), and parentheses to override the precedence of operations. Each expression can be represented by a unique binary tree whose structure is determined by the precedence of operations in the expression. Such a tree is called an expression tree.
BINARY TREES
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EXAMPLE 11.17 An Expression Tree
Figure 11.19 shows the expression tree for the expression (5 x)*y + 6/(x + z).
Here is a recursive algorithm for building an expression tree: Algorithm 11.6 Build an Expression Tree Figure 11.19 An expression tree The expression tree for a given expression can be built recursively from the following rules: 1. The expression tree for a single operand is a single root node that contains it. 2. If E 1 and E 2 are expressions represented by expression trees T 1 and T 2 , and if op is an operator, then the expression tree for the expression E 1 op E 2 is the tree with root node containing op and subtrees T 1 and T 2 . An expression has three representations, depending upon which traversal algorithm is used to traverse its tree. The preorder traversal produces the prefix representation, the inorder traversal produces the infix representation, and the postorder traversal produces the postfix representation of the expression. The postfix representation is also called reverse Polish notation or RPN. These are outlined on page 109. EXAMPLE 11.18 The Three Representations of an Expression
The three representations for the expression in Example 11.17 are: Prefix: +*-5xy/6+xz Infix: 5-x*y+6/x+z Postfix (RPN): 5x-y*6xz+/+
Ordinary function syntax uses the prefix representation. The expression in Example 11.17 could be evaluated as
sum(product(difference(5, x), y), quotient(6, sum(x, z)))
Some scientific calculators use RPN, requiring both operands to be entered before the operator. The next algorithm can be applied to a postfix expression to obtain its value. Algorithm 11.7 Evaluating an Expression from Its Postfix Representation To evaluate an expression represented in postfix, scan the representation from left to right: 1. Create a stack for operands. 2. Repeat steps 3 9 until the end of representation is reached. 3. Read the next token t from the representation. 4. If it is an operand, push its value onto the stack. 5. Otherwise, do steps 6 9: 6. Pop a from the stack. 7. Pop b from the stack. 8. Evaluate c = a t b. 9. Push c onto the stack. 10. Return the top element on the stack.
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