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Binary Search Trees (BSTs)
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What is a Binary search tree?
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Why Binary search trees?
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Binary search tree implementation
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Insertion in a BST
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Deletion from a BST
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TreeSort
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BSTs as Priority Queues
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Binary Search Tree (Definition)
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•
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A binary search tree (BST) is a binary tree that is empty or that
satisfies the BST ordering property:
1. The key of each node is greater than each key in the left
subtree, if any, of the node.
2. The key of each node is less than each key in the right subtree,
if any, of the node.
Thus, each key in a BST is unique.
A
Examples:
6
B
2
1
8
4
3
7
C
9
D
5
Note: The literature contains three other definitions for BST that allow duplicate keys in a
BST. For any node x:
• key[leftSubtree(x)]  key[x] < key[rightSubtree(x)]
• key[leftSubtree(x)] < key[x]  key[rightSubtree(x)]
• key[leftSubtree(x)]  key[x]  key[rightSubtree(x)]
We will not use any of these definitions in this course.
2
Binary Search Tree (Definition) (Contd.)
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A common misunderstanding is that the BST ordering property is only between
parents and children, rather than all the values in left and right subtrees. It is a
common error to interpret the BST ordering property as:
1. The key of each node is greater than the key of the left child of that node,
if any.
2. The key of each node is less than the key of the right child of that node, if
any
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Example:
20
10
5
40
22
12
30
45
25
The above is not a BST because both 22 and 25 cannot be on the left subtree of
20; however each node satisfies the BST property with respect to its two
children.
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Why BSTs?
1. BSTs provide good logarithmic time performance in the best and average cases.
• Average case complexities of using linear data structures compared to BSTs:
Data Structure
Retrieval or
Search
BST
O(log n)
Sorted Array
O(log n)
(using Binary
Search)
Unsorted Array
Sorted Linked List
LinkedList
Insertion
Deletion
Traversal
O(log n)
O(n)
O(n)
O(n)
O(n)
O(n)
O(1)
O(n)
O(n)
O(n)
O(n)
O(n)
O(n)
O(n)
O(1)
O(n)
O(n)
O(log n)
Note: BST worst execution time for each of the above operations is O(n) when the
tree is linear
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Why BSTs? (Contd.)
2. Binary Search Trees (BSTs) are an important data structure for dynamic sets and
Dictionaries:
Each element is an Association object having a Comparable key and an
Object value
Dynamic sets support queries such as:
Search(x), Minimum(), Maximum(), Successor(x), Predecessor(x)
They also support modifying operations like: Insert(x) and Delete(x)
3. BSTs can be used to implement priority queues
4. BSTs are used in TreeSort
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Binary Search Tree Implementation
• The BinarySearchTree class inherits the instance variables
key, left, and right of the BinaryTree class:
public class BinarySearchTree extends BinaryTree
implements SearchableContainer {
private BinarySearchTree getLeftBST(){
return (BinarySearchTree) getLeft( ) ;
}
private BinarySearchTree getRightBST( ){
return (BinarySearchTree) getRight( ) ;
}
// . . .
}
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Binary Search Tree Implementation: find
The recursive find method of the BinarySearchTree class:
public Comparable find(Comparable target)
{
if(isEmpty())
return null;
Comparable currentKey = (Comparable) key;
int comparison = target.compareTo(currentKey);
if(comparison == 0)
return currentKey;
else if(comparison < 0)
return getLeftBST().find(target);
else
return getRightBST().find(target);
}
7
Binary Search Tree Implementation: findIterative
The iterative find method of the BinarySearchTree class:
public Comparable findIterative(Comparable target){
if(isEmpty())
return null;
BinarySearchTree tree = this;
while(!tree.isEmpty()){
Comparable currentKey = (Comparable) key;
int comparison = currentKey.compareTo(target);
if(comparison == 0)
return currentKey;
else if(comparison < 0)
tree = tree.getLeftBST();
else
tree = tree.getRightBST();
}
return null;
}
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Binary Search Tree Implementation: findMin
• The findMin method of the BinarySearchTree class:
• By the BST ordering property, the minimum key is the key of the
left-most node that has an empty left-subtree.
public Comparable findMin()
{
if(isEmpty())
return null;
if(getLeftBST().isEmpty())
return (Comparable)getKey();
else
return getLeftBST().findMin();
}
20
10
4
15
30
25
40
Exercise: Write the iterative findMin
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Binary Search Tree Implementation: findMax
• The findMax method of the BinarySearchTree class:
• By the BST ordering property, the maximum key is the key of the
right-most node that has an empty right-subtree.
public Comparable findMax() {
if(isEmpty())
return null;
if(getRightBST().isEmpty())
return (Comparable)getKey();
else
return getRightBST().findMax();
}
20
10
4
15
30
25
40
Exercise: Write the iterative findMax
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Binary Search Tree Implementation: findSuccessor
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The successor of a key x is the smallest key greater than x.
If(x has a non-empty right subtree)
Successor of x is the minimum value in the right subtree of x
else{
if(x is maximum value in tree)
x has no successor
else
successor is smallest ancestor of x that is greater than x
}
Example:
20
10
4
15
30
25
40
key
successor
4
7
9
10
15
20
30
32
35
40
40
No successor
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Binary Search Tree Implementation: findSuccessor (Contd.)
public Comparable findSuccessor(Comparable x){
if(isEmpty()) throw new IllegalArgumentException("Tree is empty");
else return findSuccessor(x, (Comparable)getKey());
}
private Comparable findSuccessor(Comparable x, Comparable successor){
if(isEmpty()) throw new IllegalArgumentException(x + " is not in tree");
Comparable currentKey = (Comparable)getKey();
int comparison = x.compareTo(currentKey);
if(comparison == 0){
if(!getRightBST().isEmpty()) return getRightBST().findMin();
else if(getRightBST().isEmpty() && successor.compareTo(x) < 0) // x is max value in tree
throw new IllegalArgumentException(x + " has no successor");
else return successor;
}
else if(comparison < 0)
return getLeftBST().findSuccessor(x, currentKey);
else
return getRightBST().findSuccessor(x, successor);
}
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Binary Search Tree Implementation: findPredecessor
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The predecessor of a key x is the largest key smaller than x.
If(x has a non-empty left subtree)
predecessor of x is the maximum value in the left subtree of x
else{
if(x is minimum value in tree)
x has no predecessor
else
predecessor is the last ancestor of x that is smaller than x (on the path from
root to the node x)
}
Example:
20
10
4
15
30
25
40
key
predecessor
4
No predecessor
9
7
15
10
32
30
35
32
40
35
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Binary Search Tree Implementation: findPredecesor (Contd.)
public Comparable findPredecessor(Comparable x){
if(isEmpty()) throw new IllegalArgumentException("Tree is empty");
else if(x.equals(getKey()) && getLeftBST().isEmpty())
throw new IllegalArgumentException(x + " has no predecessor");
else return findPredecessor(x, (Comparable)getKey());
}
private Comparable findPredecessor(Comparable x, Comparable predecessor){
if(isEmpty()) throw new IllegalArgumentException(x + " is not in tree");
Comparable currentKey = (Comparable)getKey();
int comparison = x.compareTo(currentKey);
if(comparison == 0){
if(!getLeftBST().isEmpty()) return getLeftBST().findMax();
else if(getLeftBST().isEmpty() && predecessor.compareTo(x) > 0) // x is min value in tree
throw new IllegalArgumentException(x + " has no predecessor");
else return predecessor;
}
else if(comparison > 0) return getRightBST().findPredecessor(x, currentKey);
else return getLeftBST().findPredecessor(x, predecessor);
}
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Insertion in a BST
• By the BST ordering property, a new node is always inserted as a
leaf node.
• The insert method, given in the next page, recursively finds an
appropriate empty subtree to insert the new key. It then transforms
this empty subtree into a leaf node by invoking the attachKey
method:
public void attachKey(Object obj) {
if(!isEmpty())
throw new InvalidOperationException();
else {
key = obj;
left = new BinarySearchTree();
right = new BinarySearchTree();
}
}
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Insertion in a BST (Contd.)
public void insert(Comparable comparable){
if(isEmpty())
attachKey(comparable);
else {
Comparable key = (Comparable) getKey();
if(comparable.compareTo(key)==0)
throw new IllegalArgumentException("duplicate key");
else if (comparable.compareTo(key)<0)
getLeftBST().insert(comparable);
else
getRightBST().insert(comparable);
}
}
6
5
6
2
1
4
3
5
8
7
9
2
1
3
7
5
2
8
4
6
6
9
1
4
3
2
8
7
9
1
8
4
3
7
9
5
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Deletion in a BST
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There are three cases:
1. The node to be deleted is a leaf node.
2. The node to be deleted has one non-empty child.
3. The node to be deleted has two non-empty children.
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CASE 1: Deleting a Leaf Node
Convert the leaf node into an empty tree by using the detachKey method:
// In Binary Tree class
public Object detachKey( ){
if(! isLeaf( )) throw new InvalidOperationException( ) ;
else {
Object obj = key ;
key = null ;
left = null ;
right = null ;
return obj ;
}
}
•
Example:
7
2
1
15
4
3
8
6
5
Delete 5
40
9
18
CASE 2: Deleting a one-child node
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CASE 2: THE NODE TO BE DELETED HAS ONE NON-EMPTY CHILD
(a) The right subtree of the node x to be deleted is empty.
// Let target be a reference to the node x.
BinarySearchTree temp = target.getLeftBST();
target.key = temp.key;
target.left = temp.left;
target.right = temp.right;
temp = null;
•
Example:
target
20
Delete 10
10
temp
5
3
35
22
8
target
5
40
25
20
3
35
8
6
22
40
25
6
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CASE 2: Deleting a one-child node (Cont’d)
(b) The left subtree of the node x to be deleted is empty.
// Let target be a reference to the node x.
BinarySearchTree temp = target.getRightBST();
target.key = temp.key;
target.left = temp.left;
target.right = temp.right;
temp = null;
Example:
7
7
2
1
target
4
3
8
6
5
40
12
9
2
Delete 8
15
temp
14
1
15
target
4
3
12
6
9
40
14
5
20
CASE 3: DELETING A NODE THAT HAS TWO NON-EMPTY CHILDREN
DELETION BY COPYING: METHOD#1
Copy the minimum key in the right subtree of x to the node x, then
delete the one-child or leaf-node with this minimum key.
• Example:
Delete 7
7
8
2
1
2
15
4
3
8
6
5
40
9
1
15
4
3
9
40
6
5
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CASE 3: DELETING A NODE THAT HAS TWO NON-EMPTY CHILDREN (Contd.)
DELETION BY COPYING: METHOD#2
Copy the maximum key in the left subtree of x to the node x, then
delete the one-child or leaf-node with this maximum key.
•
Example:
Delete 7
7
6
2
1
2
15
4
3
8
6
40
9
1
15
4
3
8
5
40
9
5
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Deletion by Copying: Method#1 implementation
// find the minimum key in the right subtree of the target node
Comparable min = target.getRightBST().findMin();
// copy the minimum value to the target
target.key = min;
// delete the one-child or leaf node having the min
target.getRightBST().withdraw(min);
Note: All the different cases for deleting a node are handled in the
withdraw (Comparable key) method of BinarySearchTree class
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Tree Sort
Any of the following in-order traversal behaviour of BST can be used to
implement a sorting algorithm on an array of distinct elements:
• The in-order traversal of a BST visits the nodes of the tree in increasing
sorted order
• The reverse in-order traversal of a BST visits the nodes in decreasing
sorted order
This algorithm is called TreeSort.
Example:
20
10
4
15
30
25
40
In-order Traversal: 4, 7, 9, 10, 15, 20, 25, 30, 32, 35, 40
Reverse in-order Traversal: 40, 35, 32, 30, 25, 20, 15, 10, 9, 7, 4
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Tree Sort (Contd.)
TreeSort algorithm:
• Insert the array elements in a BST
• Perform an in-order traversal on the BST, storing each visited value in the
corresponding array location
public static void treeSort(int[] x){
BinarySearchTree tree = buildBST(x);
if(tree == null)
throw new IllegalArgumentException("Error: Duplicate keys");
else{
int[] index = {0}; // need to pass index by reference
tree.treeSort(x, index);
}
}
private void treeSort(int[] x, int[] index){
if(isEmpty()) return;
else{
getLeftBST().treeSort(x, index);
int k = index[0];
x[k] = (Integer) getKey();
index[0] = k + 1;
getRightBST().treeSort(x, index);
}
}
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Tree Sort (Contd.)
private static BinarySearchTree buildBST(int[ ] x){ // x must have distinct values
BinarySearchTree tree = new BinarySearchTree( );
for(int k = 0; k < x.length; k++){
try{
tree.insert(new Integer(x[k]));
}
catch(IllegalArgumentException e){
tree = null;
}
}
return tree;
}
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Tree Sort (Contd.)
An alternative TreeSort algorithm is:
• Insert the array elements in a BST
• for(int k = 0; k < array.length; k++)
array[k] = bst.Min();
bst.exractMin();
}
public static void treeSort(int[ ] x){
BinarySearchTree tree = buildBST(x);
if(tree == null)
throw new IllegalArgumentException("Error: Duplicate keys");
else{
for(int k = 0; k < x.length; k++){
Comparable min = tree.findMin();
x[k] = (Integer) min;
tree.withdraw(min);
}
}
}
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Tree Sort (Contd.)
Adding items to a binary search tree is on average an O(log n) process, so
adding n items is an O(n log n) process.
But adding an item to an unbalanced binary tree needs O(n) time in the worstcase, when the tree resembles a linked list (degenerate tree), causing a worst
case of O(n2) for this sorting algorithm.
The worst case scenario happens when Tree Sort algorithm sorts an already
sorted array. This would make the time needed to insert all elements into the
binary tree O(n2).
The worst-case behaviour can be improved upon by using a Self-balancing
binary search tree such as AVL tree. Using such a tree, the algorithm has an
O(n log n) worst-case performance.
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BSTs as Priority Queues
By using insert and withdrawMax or WithdrawMin a BST can be used as a
priority queue in which the keys of the elements are distinct.
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