binary search trees definition of binary search tree a binary tree. each node has a (key, value)...
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Binary Search Trees
Definition Of Binary Search Tree
• A binary tree.• Each node has a (key, value) pair.• For every node x, all keys in the left
subtree of x are smaller than that in x.• For every node x, all keys in the right
subtree of x are greater than that in x.
Example Binary Search Tree20
10
6
2 8
15
40
30
25
Only keys are shown.
ADT bsTree
AbstractDataType bsTree
{Instances
OperationsFind(k)
Insert(p)
Erase(k)
Ascend() // in-order tree traversal
}
The Operation ascend()20
10
6
2 8
15
40
30
25
Do an inorder traversal.
The Operation find(k)20
10
6
2 8
15
40
30
25
The Operation insert()20
10
6
2 8
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40
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25
Insert a pair whose key is 35.
35
The Operation insert()
Insert a pair whose key is 7.
20
10
6
2 8
15
40
30
25 35
7
The Operation insert()20
10
6
2 8
15
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25
Insert a pair whose key is 18.
35
7
18
The Operation insert()20
10
6
2 8
15
40
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Complexity of insert() is O(height).
35
7
18
The Operation erase()
Three cases:
Element is in a leaf.
Element is in a degree 1 node (has one child).
Element is in a degree 2 node (has two children).
Erase From A Leaf
Erase a leaf element. key = 7
20
10
6
2 8
15
40
30
25 35
7
18
Erase From A Leaf (contd.)
Erase a leaf element. key = 35
20
10
6
2 8
15
40
30
25 35
7
18
Remove node with one child
• If node n has one child, move n’s child up to take n’s place.
Erase From A Degree 1 Node
Erase from a degree 1 node. key = 15
20
10
6
2 8
15
40
30
25 35
7
18
Erase From A Degree 1 Node (contd.)
Erase from a degree 1 node. key = 15
20
10
6
2 8
18
40
30
25 35
7
Erase From A Degree 1 Node(contd.)
Erase from a degree 1 node. key = 40
20
10
6
2 8
15
40
30
25 35
7
18
Remove node with two children
• If node n has two children, let x be node in n’s right subtree with smallest key,
• Remove x• Replace n’s key with x’s key
20
10 40
Remove node with two children
• If node n has two children, let x be node in n’s left subtree with largest key,
• Remove x• Replace n’s key with x’s key
20
10 40
Erase From A Degree 2 Node
Erase from a degree 2 node. key = 10
20
10
6
2 8
15
40
30
25 35
7
18
Erase From A Degree 2 Node20
10
6
2 8
15
40
30
25
Replace with largest key in left subtree (or smallest in right subtree).
35
7
18
Erase From A Degree 2 Node20
10
6
2 8
15
40
30
25
Replace with largest key in left subtree (or smallest in right subtree).
35
7
18
Erase From A Degree 2 Node20
8
6
2 8
15
40
30
25
Replace with largest key in left subtree (or smallest in right subtree).
35
7
18
Erase From A Degree 2 Node20
8
6
2 8
15
40
30
25
Largest key must be in a leaf or degree 1 node.
35
7
18
Another Erase From A Degree 2 Node
Erase from a degree 2 node. key = 20
20
10
6
2 8
15
40
30
25 35
7
18
Erase From A Degree 2 Node20
10
6
2 8
15
40
30
25
Replace with largest in left subtree.
35
7
18
Erase From A Degree 2 Node20
10
6
2 8
15
40
30
25
Replace with largest in left subtree.
35
7
18
Erase From A Degree 2 Node18
10
6
2 8
15
40
30
25
Replace with largest in left subtree.
35
7
18
Erase From A Degree 2 Node18
10
6
2 8
15
40
30
25 35
7
Exercise
• Start with an empty binary search tree.– Insert the keys 4,12,8,16,6,18,24,2,14,3, draw
the tree following each insert.– From the tree above, delete the keys, 6,14,16,4
in order, draw the search tree following each deletion.
// abstract class bsTree
// abstract data type specification for binary search trees
// all methods are pure virtual functions
// K is key type and E is value type
#ifndef bsTree_
#define bsTree_
#include "dictionary.h"
using namespace std;
template<class K, class E>
class bsTree : public dictionary<K,E>
{
public:
virtual void ascend() = 0;
// output in ascending order of key
};
#endif
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