1 a two-level binary expression ‘-’ ‘8’ ‘5’ treeptr inorder traversal : 8 - 5 has value...
TRANSCRIPT
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A Two-Level Binary Expression
‘-’
‘8’ ‘5’
treePtr
INORDER TRAVERSAL: 8 - 5 has value 3
PREORDER TRAVERSAL: - 8 5
POSTORDER TRAVERSAL: 8 5 -
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Levels Indicate Precedence
When a binary expression tree is used to represent an expression, the levels of the nodes in the tree indicate their relative precedence of evaluation.
Operations at higher levels of the tree are evaluated later than those below them.
The operation at the root is always the last operation performed.
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A Binary Expression Tree
‘*’
‘+’
‘4’
‘3’
‘2’
What value does it have? ( 4 + 2 ) * 3 = 18
What infix, prefix, postfix expressions does it represent?
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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
5
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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Evaluate:
‘*’
‘-’
‘8’ ‘5’
What infix, prefix, postfix expressions does it represent?
‘/’
‘+’
‘4’
‘3’
‘2’
8
Answer
Infix: ( ( 8 - 5 ) * ( ( 4 + 2 ) / 3 ) )
Prefix: * - 8 5 / + 4 2 3
Postfix: 8 5 - 4 2 + 3 / * has operators in order used
‘*’
‘-’
‘8’ ‘5’
‘/’
‘+’
‘4’
‘3’
‘2’
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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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TreeNodes
struct TreeNode {
InfoNode info ; // Data member
TreeNode* left ; // Pointer to left child
TreeNode* right ; // Pointer to right child};
. left . info . right
NULL 6000
. whichType . operand
OPERAND 7
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class ExprTree
‘*’
‘+’
‘4’
‘3’
‘2’
ExprTree
~ExprTree
Build
Evaluate . . .
private:
root
int Eval ( TreeNode* ptr )
// Pre: ptr is a pointer to a binary expression tree.// Post: Function value = the value of the expression represented// by the binary tree pointed to by 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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A Full Binary Tree
A full binary tree is a binary tree in which all the leaves are on the same level and every non leaf node has two children.
SHAPE OF A FULL BINARY TREE
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A Complete Binary Tree
A complete binary tree is a binary tree that is either full or full through the next-to-last level, with the leaves on the last level as far to the left as possible.
SHAPE OF A COMPLETE BINARY TREE
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What is a Heap?
A heap is a binary tree that satisfies these
special SHAPE and ORDER properties:
Shape: a complete binary tree.
Order: the value stored in each node is greater than or equal to the value in each of its children.
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Are These Both Heaps?
C
A T
treePtr
50
20
18
30
10
17
70
60
40 30
12
8 10
tree
Is This a Heap?
18
70
60
40 30
12
8
tree
Where is the Largest Element in a Heap?
19
70
0
60
1
40
3
30
4
12
2
8
5
tree
Number the Nodes Left to Right by Level
20
70
0
60
1
40
3
30
4
12
2
8
5
tree
Use the Numbers as Array Indexes to Store the Tree
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
70
60
12
40
30
8
tree.nodes
// HEAP SPECIFICATION
// Assumes ItemType is either a built-in simple data type// or a class with overloaded realtional operators.
template< class ItemType >struct HeapType
{ void ReheapDown ( int root , int bottom ) ;
void ReheapUp ( int root, int bottom ) ;
ItemType* elements ; // ARRAY to be allocated dynamically
int numElements ;
};
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Observations
Heap is a struct with member functions
Why not a class? To allow access to its components Heaps are usually used in conjunction
with other classes which want access to the components but will hide them from the application
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Reheap Down For heaps with all but the top element
in heap position How:
Swap root with left or right child Reheap (down) with that child
Used to remove the highest element: Remove the root Put last element in root position Reheap down to restore the order
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Reheap Up
For a node that violates a heap up: swap with parent until it is a heap
So, if we added a node to the “end” of the tree, reheap up would put it in the right spot
To build a heap: start with empty heap add element to the end and reheap up
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Exercises
Make a heap out of elements
D H A K P R Remove the top element and reheap.
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// IMPLEMENTATION OF RECURSIVE HEAP MEMBER FUNCTIONS
template< class ItemType >void HeapType<ItemType>::ReheapDown ( int root, int bottom )
// Pre: root is the index of the node that may violate the heap // order property// Post: Heap order property is restored between root and bottom
{ int maxChild ; int rightChild ; int leftChild ;
leftChild = root * 2 + 1 ; rightChild = root * 2 + 2 ; 26
ReheapDown
if ( leftChild <= bottom ) // ReheapDown continued {
if ( leftChild == bottom ) maxChild = leftChild ;else{ if (elements [ leftChild ] <= elements [ rightChild ] )
maxChild = rightChild ; else
maxChild = leftChild ;}if ( elements [ root ] < elements [ maxChild ] ){ Swap ( elements [ root ] , elements [ maxChild ] ) ; ReheapDown ( maxChild, bottom ) ;}
}}
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// IMPLEMENTATION continued
template< class ItemType >void HeapType<ItemType>::ReheapUp ( int root, int bottom )
// Pre: bottom is the index of the node that may violate the heap // order property. The order property is satisfied from root to // next-to-last node.// Post: Heap order property is restored between root and bottom
{ int parent ;
if ( bottom > root ) {
parent = ( bottom - 1 ) / 2;if ( elements [ parent ] < elements [ bottom ] ){
Swap ( elements [ parent ], elements [ bottom ] ) ;ReheapUp ( root, parent ) ;
} }} 28
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Priority Queue
A priority queue is an ADT with the property that only the highest-priority element can be accessed at any time.
Example: homework assignments are ordered by priority.
Like a sorted list except for the access restriction.
Goal: be able to insert in order and to access the top element efficiently.
ADT Priority Queue Operations
Transformers MakeEmpty Enqueue Dequeue
Observers IsEmpty IsFull
change state
observe state
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// CLASS PQTYPE DEFINITION AND MEMBER FUNCTIONS //----------------------------------------------------
template<class ItemType>
class PQType {
public:
PQType( int );
~PQType ( );
void MakeEmpty( );
bool IsEmpty( ) const;
bool IsFull( ) const;
void Enqueue( ItemType item );
void Dequeue( ItemType& item );
private:
int numItems;
HeapType<ItemType> items;
int maxItems;
};
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PQType
~PQType
Enqueue
Dequeue . . .
class PQType<char>
[0] [1] [2] [3] [4] [5] [6] [7] [8] [9]
‘X’ ‘C’ ‘J’
Private Data:
numItems
3
maxItems
10
items
.elements .numElements
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Graphs
A data structure that consists of a set of nodes (vertices) and arcs (edges) that relate the nodes to each other.
G = (V, E)
E.g. a map with V = cities and E = roads
Undirected versus directed: some of the streets may be one-way.
Weighted: each edge has a value (miles).
Applications
Traversal: Find a route (path) from Boston to
Chicago.
Find the shortest path.
Approaches:Depth-first search
Breadth-first search
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Implementation viaAdjacency Matrices
For a graph with N vertices:numVertices = the number of nodes
vertices[N] = name of vertex N
edges[N] [M] = weight from vertex N to M
See p. 554. How do you add a vertex? An edge? How do you find a path?
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Implementation viaAdjacency Lists
An array of vertices. Each vertex has a linked list of the
nodes extending directly from it. See p. 559. When would you use this? When not?