1. Based on Levent Akın’s CmpE160 Lecture Slides5
Linked Structures 3
Binary Trees
Based on Levent Akın’s CmpE160 Lecture Slides

2. Jake’s Pizza Shop Owner Jake Manager Chef Brad Carol
Waitress Waiter Cook Helper
Joyce Chris Max Len

3. Definition 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.

4. A Tree Has a Root Node ROOT NODE Owner Jake Manager Chef Brad Carol
Waitress Waiter Cook Helper
Joyce Chris Max Len

5. Leaf Nodes have No Children
Owner
Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper
Joyce Chris Max Len

6. A Tree Has Leaves LEVEL 0 Owner Jake Manager Chef Brad Carol
Waitress Waiter Cook Helper
Joyce Chris Max Len

7. Level One LEVEL 1 Owner Jake Manager Chef Brad Carol
Waitress Waiter Cook Helper
Joyce Chris Max Len

8. Level Two LEVEL 2 Owner Jake Manager Chef Brad Carol
Waitress Waiter Cook Helper
Joyce Chris Max Len
LEVEL 2

9. A Subtree LEFT SUBTREE OF ROOT NODE Owner Jake Manager Chef Brad Carol
Waitress Waiter Cook Helper
Joyce Chris Max Len
LEFT SUBTREE OF ROOT NODE

10. Another Subtree RIGHT SUBTREE OF ROOT NODE Owner Jake
Manager Chef Brad Carol
Waitress Waiter Cook Helper
Joyce Chris Max Len
RIGHT SUBTREE
OF ROOT NODE

11. 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.

12. 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.

13. Sample Tree Level and Depth Level 1 2 3 4 3 1 2 2 1 2 3 2 2 3 3 3 1 33 1 3 3 3 4 4 4

14. 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

15. 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 1/2
3 (2n+1)/3n, or about 2/3
4 (3n+1)/4n, or about 3/4

16. Percentage of Null links

17. Tree Representations (continued)
Clearly, to reduce the proportion of null links, we need to reduce the degree of the tree.
An alternative representation for a tree of degree k:
Use a left child-right sibling representation.
Each node has two pointers, one to its left child and one to its right sibling. 13

18. Left Child - Right Sibling link structure
data
left child
right sibling A B C D E F G H I J M K L

19. The actual tree represented by itJ E F G H I M K L

20. Binary Tree 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.

21. A Binary Tree V Q L E T A K S

22. How many leaf nodes? V Q L T E A K S

23. How many descendants of Q?V Q L T E A K S

24. How many ancestors of K? Q V T K S A E L

25. Implementing a Binary Tree with Pointers and Dynamic DataV Q L T E A K S

26. Node Terminology for a Tree Node

27. Samples of Binary Trees
Complete Binary Tree A A A 1 B B 2 B C C
Skewed Binary Tree 3 D E F G D 4 H I E 5

28. Maximum Number of Nodes in BT
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.

29. Relations between Number of Leaf Nodes and Nodes of Degree 2
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

30. 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

31. 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

32. Examples of Different Types of Binary Trees

33. Full binary tree of depth 4
Full BT VS Complete BT
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 A B C B C D E F G D E F G I J K L M N O H H I
Complete binary tree
Full binary tree of depth 4

34. A Binary Tree and Its Array Representation

35. 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].

36. Sequential representation
[0]
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8] A B C D E F G H I
(1) space waste
(2) insertion/deletion
problem A
[0]
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8] .
[15] A B -- C D . E B A C B C D D E F G E H I

37. Linked Representation
typedef struct node *treePointer;
typedef struct node {
ItemType data;
treePointer leftChild;
treePointer rightChild; };
data
leftChild
data
rightChild
leftChild
rightChild

38. Binary Tree Traversals
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

39. Arithmetic Expression Using BT+ * E * D / C A B

40. Inorder Traversal (recursive version)
void inorder(treePointer ptr)
// inorder tree traversal {
if (ptr!=NULL) {
inorder(ptr->leftChild);
visit(ptr->data);
inorder(ptr->rightChild); }
A / B * C * D + E

41. Preorder Traversal (recursive version)
void preorder(treePointer ptr)
// preorder tree traversal {
if (ptr!=NULL) {
visit(ptr->data);
preorder(ptr->leftChild);
preorder(ptr->rightChild); }
+ * * / A B C D E

42. Postorder Traversal (recursive version)
void postorder(treePointer ptr)
// postorder tree traversal {
if (ptr!=NULL) {
postorder(ptr->leftChild);
postorder(ptr->rightChild);
visit(ptr->data); }
A B / C * D * E +

43. Iterative Inorder Traversal (using stack)
void iter_inorder(treePointer node) {
StackType<treePointer> NodeStack;
bool completed=false;
while (!completed) {
while(node!=NULL){
NodeStack.push(node); // add to stack
node=node->leftChild }
NodeStack.pop(node); // delete from stack
if (node!=NULL){
cout << node->data;
node = node->rightChild;
else completed=true;
O(n)

44. Trace Operations of Inorder Traversal

45. Level Order Traversal (using queue)
void level_order(treePointer ptr)
/* level order tree traversal */ {
QueueType<treePointer> NodeQueue;
if (ptr!=NULL) {
NodeQueue.enqueue(ptr);
for (;;) {
NodeQueue.dequeue(ptr);

46. cout << ptr->data; if (ptr->leftChild)
if (ptr!=NULL) {
cout << ptr->data;
if (ptr->leftChild)
NodeQueue.enqueue(ptr->leftChild);
if (ptr->rightChild!=NULL)
NodeQueue.enqueue(ptr->rightChild); }
else break;
+ * E * D / C A B

47. Arithmetic Expression Using BT
inorder traversal
A / B * C * D + E
infix expression
preorder traversal
+ * * / A B C D E
prefix expression
postorder traversal
A B / C * D * E +
postfix expression
level order traversal
+ * E * D / C A B + * E * D / C A B

48. A Binary Search Tree (BST) is . . .
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.

49. Shape of a binary search tree . . .
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.
‘J’

50. Inserting ‘E’ into the BST
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.
‘J’
‘E’

51. Inserting ‘F’ into the BST
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.
‘J’
‘E’
‘F’

52. Inserting ‘T’ into the BST
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.
‘J’
‘E’
‘F’
‘T’

53. Inserting ‘A’ into the BST
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.
‘J’
‘E’
‘F’
‘T’
‘A’

54. What binary search tree . . .
is obtained by inserting
the elements ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ in that order?
‘A’

55. Binary search tree . . . obtained by inserting
the elements ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ in that order.
‘A’
‘E’
‘F’
‘J’
‘T’

56. Another binary search tree
‘J’
‘E’
‘T’
‘A’
‘H’
‘M’
‘K’
‘P’
Add nodes containing these values in this order:
‘D’ ‘B’ ‘L’ ‘Q’ ‘S’ ‘V’ ‘Z’

57. Is ‘F’ in the binary search tree?
‘J’
‘E’
‘T’
‘A’
‘H’
‘M’
‘V’
‘D’
‘K’
‘Z’
‘P’
‘B’
‘L’
‘Q’
‘S’

58. Class TreeType // Assumptions: Relational operators overloaded{
public:
// Constructor, destructor, copy constructor
...
// Overloads assignment
// Observer functions
// Transformer functions
// Iterator pair
void Print(std::ofstream& outFile) const;
private:
TreeNode* root; };

59. 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;

60. 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? 60

61. Tree Recursion CountNodes Version 2
if (Left(tree) is NULL) AND (Right(tree) is NULL)
return 1
else if Left(tree) is NULL
return CountNodes(Right(tree)) + 1
else if Right(tree) is NULL
return CountNodes(Left(tree)) + 1
else return CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1
What happens when the initial tree is NULL? 61

62. Tree Recursion Can we simplify this algorithm? CountNodes Version 3
if tree is NULL
return 0
else if (Left(tree) is NULL) AND (Right(tree) is NULL)
return 1
else if Left(tree) is NULL
return CountNodes(Right(tree)) + 1
else if Right(tree) is NULL
return CountNodes(Left(tree)) + 1
else return CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1
Can we simplify this algorithm? 62

63. Tree Recursion CountNodes Version 4 if tree is NULL return 0 else
return CountNodes(Left(tree)) +
CountNodes(Right(tree)) + 1
Is that all there is?

64. // Implementation of Final Version
int 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;

65. Retrieval Operation

66. 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);

67. Retrieval Operation, cont.
else if (item > tree->info)
Retrieve(tree->right, item, found);
else {
item = tree->info;
found = true; }

68. The Insert Operation
A new node is always inserted into its appropriate position in the tree as a leaf.

69. Insertions into a Binary Search Tree

70. The recursive InsertItem operation

71. The tree parameter is a pointer within the tree

72. 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);

73. Deleting a Leaf Node

74. Deleting a Node with One Child

75. Deleting a Node with Two Children

76. 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 predecessor
Set Info(tree) to Info(predecessor)
Delete predecessor

77. 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); }

78. 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).

79. 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 }

80. 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?

81. Printing all the Nodes in Order

82. 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.

83. 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?

84. Destructor void Destroy(TreeNode*& tree); TreeType::~TreeType() {
Destroy(root); }
void Destroy(TreeNode*& tree)
if (tree != NULL)
Destroy(tree->left);
Destroy(tree->right);
delete tree;

85. 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)

86. 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); }

87. Inorder(tree) To print in alphabetical order if tree is not NULL
Inorder(Left(tree))
Visit Info(tree)
Inorder(Right(tree))
To print in alphabetical order

88. Postorder(tree) Visits leaves first (good for deletion)
if tree is not NULL
Postorder(Left(tree))
Postorder(Right(tree))
Visit Info(tree)
Visits leaves first (good for deletion)

89. Preorder(tree) Useful with binary trees (not binary search trees)
if tree is not NULL
Visit Info(tree)
Preorder(Left(tree))
Preorder(Right(tree))
Useful with binary trees
(not binary search trees)

90. Three Tree Traversals

91. Our Iteration Approach
The client program passes the ResetTree and
GetNextItem functions a parameter indicating
which of the three traversals to use
ResetTree generates a queues of node contents in the indicated order
GetNextItem processes the node contents from the appropriate queue: inQue, preQue, postQue.

92. Code for ResetTree void TreeType::ResetTree(OrderType order)
// Calls function to create a queue of the tree
// elements in the desired order. {
switch (order)
case PRE_ORDER : PreOrder(root, preQue);
break;
case IN_ORDER : InOrder(root, inQue);
case POST_ORDER: PostOrder(root, postQue); }

93. Code for GetNextItem void TreeType::GetNextItem(ItemType& item,
OrderType order,bool& finished) {
finished = false;
switch (order)
case PRE_ORDER : preQue.Dequeue(item);
if (preQue.IsEmpty())
finished = true;
break;
case IN_ORDER : inQue.Dequeue(item);
if (inQue.IsEmpty())
case POST_ORDER: postQue.Dequeue(item);
if (postQue.IsEmpty()) }

94. Iterative Versions FindNode Set nodePtr to tree Set parentPtr to NULL
Set found to false
while more elements to search AND NOT found
if item < Info(nodePtr)
Set parentPtr to nodePtr
Set nodePtr to Left(nodePtr)
else if item > Info(nodePtr)
Set nodePtr to Right(nodePtr)
else
Set found to true

95. Code for FindNode void FindNode(TreeNode* tree, ItemType item,
TreeNode*& nodePtr, TreeNode*& parentPtr) {
nodePtr = tree;
parentPtr = NULL;
bool found = false;
while (nodePtr != NULL && !found)
{ if (item < nodePtr->info)
parentPtr = nodePtr;
nodePtr = nodePtr->left; }
else if (item > nodePtr->info)
nodePtr = nodePtr->right;
else found = true;
Code for FindNode

96. 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);

97. Using function FindNode to find the insertion point

98. Using function FindNode to find the insertion point

99. Using function FindNode to find the insertion point

100. Using function FindNode to find the insertion point

101. Using function FindNode to find the insertion point

102. AttachNewNode if item < Info(parentPtr)
Set Left(parentPtr) to newNode
else
Set Right(parentPtr) to newNode

103. AttachNewNode(revised)
if parentPtr equals NULL
Set tree to newNode
else if item < Info(parentPtr)
Set Left(parentPtr) to newNode
else
Set Right(parentPtr) to newNode

104. 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; }

105. 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); }

106. PointersnodePtr and parentPtr Are External to the Tree

107. Pointer parentPtr is External to the Tree, but parentPtr-> left is an Actual Pointer in the Tree

108. A Binary Search Tree Stored in an Array with Dummy Values