1 5 linked structures based on levent akın’s cmpe160 lecture slides 3 binary trees
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5Linked Structures
Based on Levent Akın’s CmpE160 Lecture Slides
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Binary Trees
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
Jake’s Pizza Shop
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Definition of Tree A tree is a finite set of one or more nodes
such that: There is a specially designated node called
the root. The remaining nodes are partitioned into
n>=0 disjoint sets T1, ..., Tn, where each of these sets is a tree.
We call T1, ..., Tn the subtrees of the root.
Definition
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
ROOT NODE
A Tree Has a Root Node
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
LEAF NODES
Leaf Nodes have No Children
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
LEVEL 0
A Tree Has Leaves
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
LEVEL 1
Level One
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
LEVEL 2
Level Two
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
LEFT SUBTREE OF ROOT NODE
A Subtree
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Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper Joyce Chris Max Len
RIGHT SUBTREE OF ROOT NODE
Another Subtree
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Terminology
The degree of a node is the number of subtrees of the node
The node with degree 0 is a leaf or terminal node. All other nodes are nonterminals.
The degree of a tree is the maximum degree of the nodes of a tree.
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Terminology
A node that has subtrees is the parent of the roots of the subtrees.
The roots of these subtrees are the children of the node.
Children of the same parent are siblings. The ancestors of a node are all the nodes
along the path from the root to the node The height or depth of a tree is the maximum
level of any node in the tree.
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Level and Depth
K L
E F
B
G
C
M
H I J
D
A
Level
1
2
3
4
3
2 1 3
2 0 0 1 0 0
0 0 0
1
2 2 2
3 3 3 3 3 3
4 4 4
Sample Tree
Tree Representations For a tree of degree k (also called a k-ary
tree), we could define a node structure that contains a data field plus k link fields. Advantage: uniform node structure (can
insert/delete nodes without having to change node structure)
Disadvantage: many null link fields
data link 1 link 2 ... link k
Tree Representations (continued)
In general, for a tree of degree k with n nodes, we know that the total number of link fields in the tree is kn and that exactly n-1 of them are not null. Therefore the number of null links is
kn - (n-1) = (k-1)n + 1.degree proportion of links that are null
2 (n+1)/2n, or about 1/23 (2n+1)/3n, or about 2/34 (3n+1)/4n, or about 3/4
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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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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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Node Terminology for a Tree Node
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A
B
A
B
A
B C
GE
I
D
H
F
Complete Binary Tree
Skewed Binary Tree
E
C
D
1
2
3
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Samples of Binary Trees
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The maximum number of nodes on level i of a binary tree is 2i-1, i≥1.
The maximum number of nodes in a binary tree of depth k is 2k-1, k ≥ 1.
Prove by induction.
∑i=1
k
2i−1=2k−1
Maximum Number of Nodes in BT
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For any nonempty binary tree, T, if n0 is the number of leaf nodes and n2 the number of nodes of degree 2, then n0=n2+1
proof:
Let n and B denote the total number of nodes & branches in T.
Let n0, n1, n2 represent the nodes with no children, single child, and two children respectively.
n= n0+n1+n2, B+1=n, B=n1+2n2 ==> n1+2n2+1= n, n1+2n2+1= n0+n1+n2 ==> n0=n2+1
Relations between Number ofLeaf Nodes and Nodes of Degree 2
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Definitions
Full Binary Tree: A binary tree in which all of the leaves are on the same level and every nonleaf node has two children
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Definitions (cont.)
Complete Binary Tree: 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
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Examples of Different Types of Binary Trees
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A full binary tree of depth k is a binary tree of depth k having 2 -1 nodes, k ≥ 0.
A binary tree with n nodes and depth k is complete iff its nodes correspond to the nodes numbered from 1 to n in the full binary tree of depth k.
k
A
B C
GE
I
D
H
F
A
B C
GE
K
D
J
F
IH ONML
Full binary tree of depth 4Complete binary tree
Full BT VS Complete BT
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A Binary Tree and Its Array Representation
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With Array Representation
For any node tree.nodes[index] its left child is in tree.nodes[index*2 + 1]
right child is in tree.nodes[index*2 + 2]
its parent is in tree.nodes[(index – 1)/2].
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Sequential representation
AB--C------D--.E
[0][1][2][3][4][5][6][7][8].[15]
[0][1][2][3][4][5][6][7][8]
ABCDEFGHI
A
B
E
C
D
A
B C
GE
I
D
H
F
(1) space waste (2) insertion/deletion problem
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typedef struct node *treePointer;
typedef struct node {
ItemType data;
treePointer leftChild;
treePointer rightChild;
};
dataleftChild rightChild
data
leftChild rightChild
Linked Representation
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Let L, V, and R stand for moving left, visiting
the node, and moving right. There are six possible combinations of
traversal LVR, LRV, VLR, VRL, RVL, RLV
Adopt convention that we traverse left before right, only 3 traversals remain LVR, LRV, VLR inorder, postorder, preorder
Binary Tree Traversals
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+
*
A
*
/
E
D
C
B
Arithmetic Expression Using BT
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void inorder(treePointer ptr)
// inorder tree traversal
{
if (ptr!=NULL) {
inorder(ptr->leftChild);
visit(ptr->data);
inorder(ptr->rightChild);
}
}
A / B * C * D + E
Inorder Traversal (recursive version)
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void preorder(treePointer ptr)
// preorder tree traversal
{
if (ptr!=NULL) {
visit(ptr->data);
preorder(ptr->leftChild);
preorder(ptr->rightChild);
}
}
+ * * / A B C D E
Preorder Traversal (recursive version)
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void postorder(treePointer ptr)
// postorder tree traversal
{
if (ptr!=NULL) {
postorder(ptr->leftChild);
postorder(ptr->rightChild);
visit(ptr->data);
}
}
A B / C * D * E +
Postorder Traversal (recursive version)
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+
*
A
*
/
E
D
C
B
inorder traversalA / B * C * D + Einfix expressionpreorder traversal+ * * / A B C D Eprefix expressionpostorder traversalA B / C * D * E +postfix expressionlevel order traversal+ * E * D / C A B
Arithmetic Expression Using BT
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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’
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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’
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Class TreeType // Assumptions: Relational operators overloaded class TreeType { public: // Constructor, destructor, copy constructor ... // Overloads assignment ... // Observer functions ... // Transformer functions ... // Iterator pair ... void Print(std::ofstream& outFile) const; private: TreeNode* root; };
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bool TreeType::IsFull() const{ NodeType* location; try { location = new NodeType; delete location; return false; } catch(std::bad_alloc exception) { return true; }}
bool TreeType::IsEmpty() const{ return root == NULL;}
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Tree Recursion
CountNodes Version 1 if (Left(tree) is NULL) AND (Right(tree) is
NULL)return 1
else return CountNodes(Left(tree)) +
CountNodes(Right(tree)) + 1
What happens when Left(tree) is NULL?
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Tree Recursion
CountNodes Version 2if (Left(tree) is NULL) AND (Right(tree) is
NULL) return 1else if Left(tree) is NULL
return CountNodes(Right(tree)) + 1else if Right(tree) is NULL
return CountNodes(Left(tree)) + 1else return CountNodes(Left(tree)) +
CountNodes(Right(tree)) + 1
What happens when the initial tree is NULL?
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Tree Recursion
CountNodes Version 3if tree is NULL
return 0else if (Left(tree) is NULL) AND (Right(tree)
is NULL) return 1else if Left(tree) is NULL
return CountNodes(Right(tree)) + 1else if Right(tree) is NULL
return CountNodes(Left(tree)) + 1else return CountNodes(Left(tree)) +
CountNodes(Right(tree)) + 1
Can we simplify this algorithm?
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Tree Recursion
CountNodes Version 4if tree is NULL
return 0else
return CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1
Is that all there is?
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// Implementation of Final Versionint CountNodes(TreeNode* tree); // Pototype int TreeType::LengthIs() const// Class member function{ return CountNodes(root);} int CountNodes(TreeNode* tree)// Recursive function that counts the nodes{ if (tree == NULL) return 0; else return CountNodes(tree->left) + CountNodes(tree->right) + 1;}
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Retrieval Operation
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Retrieval Operation
void TreeType::RetrieveItem(ItemType& item, bool& found)
{
Retrieve(root, item, found);
}
void Retrieve(TreeNode* tree,
ItemType& item, bool& found)
{
if (tree == NULL)
found = false;
else if (item < tree->info)
Retrieve(tree->left, item, found);
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Retrieval Operation, cont.
else if (item > tree->info)
Retrieve(tree->right, item, found);
else
{
item = tree->info;
found = true;
}
}
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The Insert Operation
A new node is always inserted into its appropriate position in the tree as a leaf.
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Insertions into a Binary Search Tree
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The recursive InsertItem operation
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Recursive Insert
void Insert(TreeNode*& tree, ItemType item){ if (tree == NULL) {// Insertion place found. tree = new TreeNode; tree->right = NULL; tree->left = NULL; tree->info = item; } else if (item < tree->info) Insert(tree->left, item); else Insert(tree->right, item); }
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Deleting a Leaf Node
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Deleting a Node with One Child
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Deleting a Node with Two Children
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DeleteNode Algorithm
if (Left(tree) is NULL) AND (Right(tree) is NULL)Set tree to NULL
else if Left(tree) is NULL Set tree to Right(tree)else if Right(tree) is NULL
Set tree to Left(tree)else
Find predecessorSet Info(tree) to Info(predecessor)Delete predecessor
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Code for DeleteNode
void DeleteNode(TreeNode*& tree){ ItemType data; TreeNode* tempPtr; tempPtr = tree; if (tree->left == NULL) { tree = tree->right; delete tempPtr; } else if (tree->right == NULL){ tree = tree->left; delete tempPtr;} else { GetPredecessor(tree->left, data); tree->info = data; Delete(tree->left, data); }}
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Definition of Recursive Delete
Definition: Removes item from tree Size: The number of nodes in the path from the
root to the node to be deleted. Base Case: If item's key matches key in
Info(tree), delete node pointed to by tree.General Case: If item < Info(tree),
Delete(Left(tree), item); else
Delete(Right(tree), item).
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Code for Recursive Delete
void Delete(TreeNode*& tree, ItemType item)
{ if (item < tree->info) Delete(tree->left, item); else if (item > tree->info) Delete(tree->right, item); else DeleteNode(tree); // Node found}
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Code for GetPredecessor
void GetPredecessor(TreeNode* tree,
ItemType& data)
{
while (tree->right != NULL)
tree = tree->right;
data = tree->info;
}
Why is the code not recursive?
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Printing all the Nodes in Order
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Function Print
Function Print Definition: Prints the items in the binary search
tree in order from smallest to largest. Size: The number of nodes in the tree
whose root is tree Base Case: If tree = NULL, do nothing. General Case: Traverse the left subtree in order.
Then print Info(tree). Then traverse the right subtree in order.
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Code for Recursive InOrder Print
void PrintTree(TreeNode* tree, std::ofstream& outFile){ if (tree != NULL) { PrintTree(tree->left, outFile); outFile << tree->info; PrintTree(tree->right, outFile); }}
Is that all there is?
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Destructor
void Destroy(TreeNode*& tree);TreeType::~TreeType(){ Destroy(root);} void Destroy(TreeNode*& tree){ if (tree != NULL) { Destroy(tree->left); Destroy(tree->right); delete tree; }}
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Algorithm for Copying a Tree
if (originalTree is NULL)
Set copy to NULL
else
Set Info(copy) to Info(originalTree)
Set Left(copy) to Left(originalTree)
Set Right(copy) to Right(originalTree)
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Code for CopyTree
void CopyTree(TreeNode*& copy, const TreeNode* originalTree){ if (originalTree == NULL) copy = NULL; else { copy = new TreeNode; copy->info = originalTree->info; CopyTree(copy->left, originalTree->left); CopyTree(copy->right, originalTree->right); }}
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Inorder(tree)
if tree is not NULLInorder(Left(tree))Visit Info(tree)Inorder(Right(tree))
To print in alphabetical order
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Postorder(tree)
if tree is not NULLPostorder(Left(tree))Postorder(Right(tree))Visit Info(tree)
Visits leaves first
(good for deletion)
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Preorder(tree)
if tree is not NULLVisit Info(tree)Preorder(Left(tree))Preorder(Right(tree))
Useful with binary trees(not binary search trees)
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Three Tree Traversals
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InsertItem
Create a node to contain the new item.
Find the insertion place.
Attach new node.
Find the insertion place
FindNode(tree, item, nodePtr, parentPtr);
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Using function FindNode to find the insertion point
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Using function FindNode to find the insertion point
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Using function FindNode to find the insertion point
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Using function FindNode to find the insertion point
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Using function FindNode to find the insertion point
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AttachNewNode
if item < Info(parentPtr) Set Left(parentPtr) to newNodeelse
Set Right(parentPtr) to newNode
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AttachNewNode(revised)
if parentPtr equals NULLSet tree to newNode
else if item < Info(parentPtr) Set Left(parentPtr) to newNodeelse
Set Right(parentPtr) to newNode
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Code for InsertItem
void TreeType::InsertItem(ItemType item){ TreeNode* newNode; TreeNode* nodePtr; TreeNode* parentPtr; newNode = new TreeNode; newNode->info = item; newNode->left = NULL; newNode->right = NULL; FindNode(root, item, nodePtr, parentPtr); if (parentPtr == NULL) root = newNode; else if (item < parentPtr->info) parentPtr->left = newNode; else parentPtr->right = newNode;}
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Code for DeleteItem
void TreeType::DeleteItem(ItemType item){ TreeNode* nodePtr; TreeNode* parentPtr; FindNode(root, item, nodePtr, parentPtr);
if (nodePtr == root) DeleteNode(root); else if (parentPtr->left == nodePtr) DeleteNode(parentPtr->left); else DeleteNode(parentPtr->right);}
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PointersnodePtr and parentPtr Are External to the Tree
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Pointer parentPtr is External to the Tree, but parentPtr-> left is an Actual Pointer in the Tree
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A Binary Search Tree Stored in an Array with Dummy Values