Post on 18-Jan-2016
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Nell Dale
Chapter 8
Binary Search Trees
Modified from the slides by Sylvia Sorkin, Community College of Baltimore County - Essex Campus
C++ Plus Data Structures
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for an element in a sorted list stored sequentially
in an array: O(Log2N) in a linked list: ? (midpoint = ?)
Binary search
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Introduce some basic tree vocabulary
Develop algorithms
Implement operations needed to use a binary search tree
Goals of this chapter
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Jake’s Pizza Shop
Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
A Tree Has a Root Node
ROOT NODE
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
Leaf nodes have no children
LEAF NODES
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
A Tree Has Levels
LEVEL 0
Level of a node: its distance from the root
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
Level One
LEVEL 1
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
Level Two
LEVEL 2
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Maximum number of levels: N
Minimum number of levels: logN + 1
e.g., N=8, 16, …, 1000
A binary tree with N nodes:
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the maximum level in a tree
The height of a tree:
-- a critical factor in determining how efficiently we can search for elements
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
A Subtree
LEFT SUBTREE OF ROOT NODE
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
Another Subtree
RIGHT SUBTREE OF ROOT NODE
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A binary tree is a structure in which:
Each node can have at most two children, and in which a unique path exists from the root to every other node.
The two children of a node are called the left child and the right child, if they exist.
Binary Tree
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A Binary Tree
Q
V
T
K S
A E
L
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A Binary Tree ?
Q
V
T
K S
A E
L
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How many leaf nodes?
Q
V
T
K S
A E
L
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How many descendants of Q?
Q
V
T
K S
A E
L
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How many ancestors of K?
Q
V
T
K S
A E
L
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Implementing a Binary Tree with Pointers and Dynamic Data
Q
V
T
K S
A E
L
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Each node contains two pointers
template< class ItemType >struct TreeNode {
ItemType info; // Data member
TreeNode<ItemType>* left; // Pointer to left child
TreeNode<ItemType>* right; // Pointer to right child
};
. left . info . right
NULL ‘A’ 6000
// BINARY SEARCH TREE SPECIFICATION
template< class ItemType >
class TreeType { public:
TreeType ( ); // constructor ~TreeType ( ); // destructor bool IsEmpty ( ) const; bool IsFull ( ) const; int NumberOfNodes ( ) const; void InsertItem ( ItemType item ); void DeleteItem (ItemType item ); void RetrieveItem ( ItemType& item, bool& found ); void PrintTree (ofstream& outFile) const;
. . .
private:
TreeNode<ItemType>* root;}; 22
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TreeType<char> CharBST;
‘J’
‘E’
‘A’
‘S’
‘H’
TreeType
~TreeType
IsEmpty
InsertItem
Private data:
root
RetrieveItem
PrintTree . . .
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A Binary Tree
Q
V
T
K S
A E
L
Search for ‘S’?
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A special kind of binary tree in which:
1. Each node contains a distinct data value,
2. The key values in the tree can be compared using “greater than” and “less than”, and
3. The key value of each node in the tree is
less than every key value in its right subtree, and greater than every key value in its left subtree.
A Binary Search Tree (BST) is . . .
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Depends on its key values and their order of insertion.
Insert the elements ‘J’ ‘E’ ‘F’ ‘T’ ‘A’ in that order.
The first value to be inserted is put into the root node.
Shape of a binary search tree . . .
‘J’
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Thereafter, each value to be inserted begins by comparing itself to the value in the root node, moving left it is less, or moving right if it is greater. This continues at each level until it can be inserted as a new leaf.
Inserting ‘E’ into the BST
‘J’
‘E’
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Begin by comparing ‘F’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf.
Inserting ‘F’ into the BST
‘J’
‘E’
‘F’
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Begin by comparing ‘T’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf.
Inserting ‘T’ into the BST
‘J’
‘E’
‘F’
‘T’
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Begin by comparing ‘A’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf.
Inserting ‘A’ into the BST
‘J’
‘E’
‘F’
‘T’
‘A’
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is obtained by inserting
the elements ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ in that order?
What binary search tree . . .
‘A’
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obtained by inserting
the elements ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ in that order.
Binary search tree . . .
‘A’
‘E’
‘F’
‘J’
‘T’A “degenerate” tree!
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Another binary search tree
Add nodes containing these values in this order:
‘D’ ‘B’ ‘L’ ‘Q’ ‘S’ ‘V’ ‘Z’
‘J’
‘E’
‘A’ ‘H’
‘T’
‘M’
‘K’ ‘P’
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Is ‘F’ in the binary search tree?
‘J’
‘E’
‘A’ ‘H’
‘T’
‘M’
‘K’
‘V’
‘P’ ‘Z’‘D’
‘Q’‘L’‘B’
‘S’
// BINARY SEARCH TREE SPECIFICATION
template< class ItemType >
class TreeType { public:
TreeType ( ) ; // constructor ~TreeType ( ) ; // destructor bool IsEmpty ( ) const ; bool IsFull ( ) const ; int NumberOfNodes ( ) const ; void InsertItem ( ItemType item ) ; void DeleteItem (ItemType item ) ; void RetrieveItem ( ItemType& item , bool& found ) ; void PrintTree (ofstream& outFile) const ;
. . .
private:
TreeNode<ItemType>* root ;}; 35
// SPECIFICATION (continued) // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -// RECURSIVE PARTNERS OF MEMBER FUNCTIONS
template< class ItemType >void PrintHelper ( TreeNode<ItemType>* ptr,
ofstream& outFile ) ;
template< class ItemType > void InsertHelper ( TreeNode<ItemType>*& ptr,
ItemType item ) ;
template< class ItemType > void RetrieveHelper ( TreeNode<ItemType>* ptr,
ItemType& item, bool& found ) ;
template< class ItemType > void DestroyHelper ( TreeNode<ItemType>* ptr ) ;
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// BINARY SEARCH TREE IMPLEMENTATION // OF MEMBER FUNCTIONS AND THEIR HELPER FUNCTIONS
// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - template< class ItemType > TreeType<ItemType> :: TreeType ( ) // constructor { root = NULL ;}
// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -template< class ItemType >bool TreeType<ItemType> :: IsEmpty( ) const{ return ( root == NULL ) ;}
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template< class ItemType > void TreeType<ItemType> :: RetrieveItem ( ItemType& item,
bool& found ){ RetrieveHelper ( root, item, found ) ; }
template< class ItemType > void RetrieveHelper ( TreeNode<ItemType>* ptr, ItemType& item,
bool& found){ if ( ptr == NULL )
found = false ; else if ( item < ptr->info ) // GO LEFT
RetrieveHelper( ptr->left , item, found ) ; else if ( item > ptr->info ) // GO RIGHT
RetrieveHelper( ptr->right , item, found ) ; else { // item = ptr->info ;
found = true ; }}
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template< class ItemType, class KF > void TreeType<ItemType, KF> :: RetrieveItem ( ItemType& item,
KF key, bool& found )// Searches for an element with the same key as item’s key. // If found, store it into item{ RetrieveHelper ( root, item, key, found ) ; }template< class ItemType > void RetrieveHelper ( TreeNode<ItemType,KF>* ptr, ItemType& item,
KF key, bool& found){ if ( ptr == NULL )
found = false ; else if ( key < ptr->info.key() ) // GO LEFT
RetrieveHelper( ptr->left , item, key, found ) ; else if ( key > ptr->info.key() ) // GO RIGHT
RetrieveHelper( ptr->right , item, key, found ) ; else { item = ptr->info ; found = true ; }}
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template< class ItemType > void TreeType<ItemType> :: InsertItem ( ItemType item ){
InsertHelper ( root, item ) ; }
template< class ItemType > void InsertHelper ( TreeNode<ItemType>*& ptr, ItemType item ){ if ( ptr == NULL ) { // INSERT item HERE AS LEAF
ptr = new TreeNode<ItemType> ;ptr->right = NULL ;ptr->left = NULL ;ptr->info = item ;
} else if ( item < ptr->info ) // GO LEFT
InsertHelper( ptr->left , item ) ; else if ( item > ptr->info ) // GO RIGHT
InsertHelper( ptr->right , item ) ;} 40
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Find the node in the tree Delete the node from the tree Three cases:
1. Deleting a leaf2. Deleting a node with only one child3. Deleting a node with two children
Note: the tree must remain a binary tree and the search property must remain intact
Delete()
template< class ItemType > void TreeType<ItemType> :: DeleteItem ( ItemType& item ){ DeleteHelper ( root, item ) ; }
template< class ItemType > void DeleteHelper ( TreeNode<ItemType>* ptr, ItemType& item){ if ( item < ptr->info ) // GO LEFT
DeleteHelper( ptr->left , item ) ; else if ( item > ptr->info ) // GO RIGHT
DeleteHelper( ptr->right , item ) ; else {
DeleteNode(ptr); // Node is found: call DeleteNode }}
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template< class ItemType > void DeleteNode ( TreeNode<ItemType>*& tree ){ ItemType data; TreeNode< ItemType>* tempPtr;
tempPtr = tree; if ( tree->left == NULL ) { tree = tree->right; delete tempPtr; } else if (tree->right == NULL ) { tree = tree->left; delete tempPtr; } else { // have two children GetPredecessor(tree->left, item); tree->info = item; DeleteHelper(tree->left, item); // Delete predecessor node (rec) }}
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template< class ItemType > void GetPredecessor( TreeNode<ItemType>* tree, ItemType& data )// Sets data to the info member of the rightmost node in tree{ while (tree->right != NULL) tree = tree->right;
data = tree->info;}
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Traverse a list:
-- forward
-- backward
Traverse a tree:
-- there are many ways!
PrintTree()
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Inorder Traversal: A E H J M T Y
‘J’
‘E’
‘A’ ‘H’
‘T’
‘M’ ‘Y’
tree
Print left subtree first Print right subtree last
Print second
// INORDER TRAVERSALtemplate< class ItemType >void TreeType<ItemType> :: PrintTree ( ofstream& outFile ) const{
PrintHelper ( root, outFile ) ; }
template< class ItemType > void PrintHelper ( TreeNode<ItemType>* ptr, ofstream& outFile ){ if ( ptr != NULL ) {
PrintHelper( ptr->left , outFile ) ; // Print left subtree
outFile << ptr->info ;
PrintHelper( ptr->right, outFile ) ; // Print right subtree
}} 47
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Preorder Traversal: J E A H T M Y
‘J’
‘E’
‘A’ ‘H’
‘T’
‘M’ ‘Y’
tree
Print left subtree second Print right subtree last
Print first
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‘J’
‘E’
‘A’ ‘H’
‘T’
‘M’ ‘Y’
tree
Print left subtree first Print right subtree second
Print last
Postorder Traversal: A H E M Y T J
template< class ItemType > TreeType<ItemType> :: ~TreeType ( ) // DESTRUCTOR{ DestroyHelper ( root ) ; }
template< class ItemType > void DestroyHelper ( TreeNode<ItemType>* ptr )// Post: All nodes of the tree pointed to by ptr are deallocated.
{ if ( ptr != NULL ) {
DestroyHelper ( ptr->left ) ;DestroyHelper ( ptr->right ) ;delete ptr ;
}}
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Read Text …
Iterative Insertion and Deletion
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Is the depth of recursion relatively shallow?
Yes. Is the recursive solution shorter or clearer than the
nonrecursive version?
Yes. Is the recursive version much less efficient than the
nonrecursive version?
No.
Recursion or Iteration?Assume: the tree is well balanced.
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Use a recursive solution when (Chpt. 7):
The depth of recursive calls is relatively “shallow” compared to the size of the problem.
The recursive version does about the same amount of work as the nonrecursive version.
The recursive version is shorter and simpler than the nonrecursive solution.
SHALLOW DEPTH EFFICIENCY CLARITY
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BST:
Quick random-access with the flexibility of a linked structure
Can be implemented elegantly and concisely using recursion
Takes up more memory space than a singly linked list
Algorithms are more complicated
Binary Search Trees (BSTs) vs. Linear Lists
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End