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## Binary Search Tree

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**1. **Binary Search Tree BST Property
All elements stored in the left subtree of a node whose value is K have values less than K. All elements stored in the right subtree of a node whose value is K have values greater than or equal to K.
That is, a nodes left child must have a key less than its parent, and a nodes right child must have a key greater than its parent.

**2. **Binary Search Tree

**3. **Binary Search Tree
Operations on BST ADT
Create a BST
Insert an element
Remove an element
Find an element
Clear (remove all elements)
Display all elements in a sorted order

**4. **Binary Search Tree Inorder traversal
Visit the left subtree, then the node, then the right subtree.
Algorithm:
If there is a left child visit it
Visit the node itself
If there is a right child visit it

**5. **Binary Search Tree Postorder traversal
Visit each node after visiting its children.
Algorithm:
If there is a left child visit it
If there is a right child visit it
Visit the node itself

**6. **Binary Search Tree Preorder traversal
Visit each node then visit its children.
Algorithm:
Visit the node itself
If there is a left child visit it
If there is a right child visit it

**7. **Binary Search Tree Insert Algorithm
If value we want to insert < key of current node, we have to go to the left subtree
Otherwise we have to go to the right subtree
If the current node is empty (not existing) create a node with the value we are inserting and place it here.

**8. **Binary Search Tree For example, inserting 15 into the BST?

**9. **Binary Search Tree Delete Algorithm
How do we delete a node form BST?
Similar to the insert function, after deletion of a node, the property of the BST must be maintained.

**10. **Binary Search Tree There are 3 possible cases
Node to be deleted has no children
? We just delete the node.
Node to be deleted has only one child
? Replace the node with its child and make the parent of the
deleted node to be a parent of the child of the deleted node
Node to be deleted has two children

**11. **Binary Search Tree Node to be deleted has two children

**12. **Binary Search Tree Node to be deleted has two children
Steps:
Find minimum value of right subtree
Delete minimum node of right subtree but keep its value
Replace the value of the node to be deleted by the minimum value whose node was deleted earlier.

**13. **Binary Search Tree

**14. **Binary Search Tree template <class DataElem>
class TreeNode
{
private:
TreeNode *leftPtr; // pointer to left subtree
DataElem data; // Node value
TreeNode *rightPtr; // pointer to right subtree
public:
TreeNode( const DataElem &d )
{ leftPtr = 0; data = d; rightPtr= 0; }
DataElem getData() const { return data; }
void setValue(DataElem val) { data = val; }
};

**15. **Binary Search Tree class BSTree
{
private:
TreeNode *rootPtr;
// utility functions
void insertNodeHelper (TreeNode **, const DataElem & );
void preOrderHelper ( TreeNode * ) const;
void inOrderHelper ( TreeNode * ) const;
void postOrderHelper ( TreeNode * ) const;
void deleteNodeHelper (TreeNode *&, DataElem);

**16. **Binary Search Tree
public:
BSTree () { rootPtr = NULL; };
void insertNode ( const DataElem & );
void preOrderTraversal () const;
void inOrderTraversal () const;
void postOrderTraversal () const;
void removeMin();
void deleteNode(DataElem);
TreeNode* deleteMin(TreeNode *&);
};

**17. **Binary Search Tree void Tree::insertNode( const DataElem &value )
{
insertNodeHelper( &rootPtr, value );
}
void Tree::deleteNode(DataElem val)
{
deleteNodeHelper( rootPtr, val );
}

**18. **Binary Search Tree // This function receives a POINTER to a pointer so the POINTER can be modified.
void Tree::insertNodeHelper( TreeNode **ptr,
const DataElem &value )
{
if ( *ptr == NULL ) // tree is empty
{
*ptr = new TreeNode( value );
assert( *ptr != NULL );
}
else // tree is not empty

**19. **Binary Search Tree // Else tree is not empty
if ( value < ( *ptr )->data )
insertNodeHelper( &( (*ptr)->leftPtr ), value );
else
if ( value > ( *ptr )->data )
insertNodeHelper( &( ( *ptr )->rightPtr ), value );
else
cout << value << " is already in tree" << endl;
}

**20. **Binary Search Tree void Tree::preOrderTraversal() const
{
preOrderHelper( rootPtr );
}
void Tree::preOrderHelper(TreeNode *ptr ) const
{
if ( ptr != NULL )
{
cout << ptr->data << ' ';
preOrderHelper( ptr->leftPtr );
preOrderHelper( ptr->rightPtr );
}
}

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