Binary Expression Trees
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Transcript of Binary Expression Trees
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CS308
Data Structures
An application of binary trees:
Binary Expression Trees
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A special kind of binary tree in which:
1. Each leaf node contains a single operand
2. Each nonleaf node contains asingle binaryoperator
3. The left and right subtrees of an operator
node represent subexpressions that mustbe evaluated beforeapplying the operator at
the root of the subtree.
A Binary Expression Tree is . . .
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A Four-Level Binary Expression
*
-
8 5
/
+
4
3
2
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Levels Indicate Precedence
The levels of the nodes in the tree indicate
their relative precedence of evaluation (wedo not need parentheses to indicate precedence).
Operations at higher levels of the tree areevaluated later than those below them.
The operation at the root is always the lastoperation performed.
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A Binary Expression Tree
What value does it have?
( 4 + 2 ) * 3 = 18
*
+
4
3
2
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Easy to generate the infix, prefix,
postfix expressions (how?)
Infix: ( ( 8 - 5 ) * ( ( 4 + 2 ) / 3 ) )
Prefix: * - 8 5 / + 4 2 3
Postfix: 8 5 - 4 2 + 3 / *
*
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8 5
/
+
4
3
2
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Inorder Traversal: (A + H) / (M - Y)
/
+
A H
-
M Y
tree
Print left subtree first Print right subtree last
Print second
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Preorder Traversal: / + A H - M Y
/
+
A H
-
M Y
tree
Print left subtree second Print right subtree last
Print first
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/
+
A H
-
M Y
tree
Print left subtree first Print right subtree second
Print last
Postorder Traversal: A H + M Y - /
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class ExprTree
*
+
4
3
2
ExprTree
~ExprTree
Build
Evaluate.
.
.
private:
TreeNode*
root;
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Each node contains two pointers
struct TreeNode
{
InfoNode info ; // Data member
TreeNode* left ; // Po in ter to left ch i ldTreeNode* right ; // Poin ter to r igh t ch ild
};
left info right
NULL 6000whichType operand
OPERAND 7
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InfoNode has 2 forms
enum OpType { OPERATOR, OPERAND } ;
struct InfoNode
{OpType whichType;
union // ANONYMOUS union
{
char operation ;
int operand ;
}};
whichType operation
OPERATOR +
whichType operand
OPERAND 7
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int Eval ( TreeNode* ptr )
{ switch ( ptr->info.whichType )
{
case OPERAND : return ptr->info.operand ;
case OPERATOR :
switch ( tree->info.operation )
{
case + : return ( Eval ( ptr->left ) + Eval ( ptr->right ) ) ;
case - : return ( Eval ( ptr->left ) - Eval ( ptr->right ) ) ;
case * : return ( Eval ( ptr->left ) * Eval ( ptr->right ) ) ;
case / : return ( Eval ( ptr->left ) / Eval ( ptr->right ) ) ;}
}
}
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Building a Binary Expression Tree from an
expression in prefix notation
Insert new nodes, each time moving to the left
until an operand has been inserted.
Backtrack to the last operator, and put the
next node to its right.
Continue in the same pattern.
