Trees & Tree Traversals

CSE 2215 - Lecture 10 – Summer 2023

Instructor : Fahmid Al Rifat, Lecturer, Dept. of CSE , UIU

fahmid@cse.uiu.ac.bd 1
What is a Tree?
• A connected acyclic graph is a tree

• In computer science, a tree is an abstract model

of a hierarchical structure

• A tree consists of nodes with

a parent-child relationship

• Applications:
 Organizational charts

 File systems

 Programming environments
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What is a Tree?

o A connected acyclic graph is a tree.

o A tree T is a set of nodes in a parent-child

relationship with the following properties:

 T has a special node r, called the root of T, with no parent

node
 Each node v of T, different from r, has a unique parent node u

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Tree Terminology

• Root: node without parent (A)

• Internal node: node with at least
one child (A, B, C, F)
• External node (Leaf ): node without
children (E, I, J, K, G, H, D)
• Ancestors of a node: parent,
grandparent, grand-grandparent,
etc.
• Descendants of a node: child,
grandchild, grand-grandchild, etc.
•Subtree: tree consisting of a node
and its descendants
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Depth

• The depth of a node v can be recursively defined as follows

 If v is the root, then the depth of v is 0.

 Otherwise, the depth of v is one plus the depth of the parent of v

Algorithm
depth(T, v)
if T.isRoot(v) then
return 0
else
return 1 + depth(T,
T.parent(v))
• Running time: O(1 + dv), dv is 5
Height

• The height of a node v can be recursively defined as follows

 If v is a leaf node, then the height of v is 0.

 Otherwise, the height of v is one plus the maximum height of a child of v

• The height of a tree T is the height of the root of T

• The height of a tree T is equal to the maximum depth of a leaf node of T

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Height

Algorithm height(T, v)
if T.isLeaf(v) then
return 0
else
h=0
for each w εT.children(v)
do
h = max(h, height(T, w))
return 1 + h

• Running time: O(Σv ε T (1 + cv)), where cv is the

number of children of v in T 7
Ordered Trees

•A tree is ordered if there is a linear ordering defined for each child of each node.
•A binary tree is an ordered tree in which every node has at most two
children.
•If each node of a tree has either zero or two children, the tree is called a proper
binary tree.

Binary Tree Proper Binary Tree

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Traversal of Trees

• A traversal of a tree T is a systematic way of visiting all the nodes of

T
• Traversing a tree involves visiting the root and traversing its subtrees
• There are the following traversal methods:
 Preorder Traversal

 Postorder Traversal

 Inorder Traversal (of a binary tree)

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Preorder Traversal (General Tree)

• In a preorder traversal, a node is visited before its descendants

• If a tree is ordered, then the subtrees are traversed according to
the order of the children

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Preorder Traversal (Binary Tree)

Traversing a binary tree in Preorder

1. Visit the root.
2. Traverse the left subtree in Preorder.
3. Traverse the right subtree in Preorder.

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Postorder Traversal (General Tree)

• In a postorder traversal, a node is visited after its descendants

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Postorder Traversal (Binary Tree)

Traversing a binary tree in Postorder

1. Traverse the left subtree in Preorder.
2. Traverse the right subtree in Preorder.
3. Visit the root.

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Inorder Traversal

• In a postorder traversal, a node is visited after its descendants

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Inorder Traversal

Traversing a binary tree in inorder

1. Traverse the left subtree in inorder.
2. Visit the root.
3. Traverse the right subtree in inorder.

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Preorder, Inorder, Postorder Traversals: Example

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Construct a Binary Tree only from Preorder,
Inorder or Postorder
• Preoder: f →b →g →i →h

• No Unique Tree if only one sequence is specified.

• Need at least two traversal sequences to construct a unique tree.

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Construct a Binary Tree from Preorder & Inorder

• Preoder: f → b → a → d → c → e → g → i → h
• Inorder: a → b → c → d → e → f → i → g → h

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Construct a Binary Tree from Postorder & Inorder

• Postoder: a → c → e → d → b → i → h → g → f
• Inorder: a → b → c → d → e → f → i → g → h

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Construct a Binary Tree from Preorder & Postorder

• Preoder: f → b → a → d → c → e → g → i → h
• Postoder: a → c → e → d → b → i → h → g → f

• No unique Tree!!
• Need at least one Inorder traversal sequence to construct Unique tree.

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Level Order Traversal

•In a level order traversal, every node on a level is visited before going to
a lower level

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Euler Tour Traversal

• Generic traversal of a binary tree

• Includes a special cases the preorder, postorder and inorder
traversals
• Walk around the tree and visit each node three times:
 on the left (preorder)

 from below (inorder)

 on the right (postorder)

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Euler Tour Traversal

• Generic traversal of a binary tree

• Includes a special cases the preorder, postorder and inorder
traversals
• Walk around the tree and visit each node three times:
 on the left (preorder)

 from below (inorder)

 on the right (postorder)

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(Proper) Binary Tree
• A (proper) binary tree is a tree with the following properties:
 Each internal node has two children
 The children of a node are an ordered pair

• We call the children of an internal node left child and right

child
• Alternative recursive definition:
• A (proper) binary tree is either
 a tree consisting of a single node, or
 a tree whose root has an ordered pair of children, each of which is a proper binary tree

Application:
 arithmetic expressions

 decision processes

 searching
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Arithmetic Expression Tree

• Binary tree associated with an arithmetic expression

 internal nodes: operators

 external nodes: operands

• Example: arithmetic expression tree for the expression (2 * (a - 1) + (3 * b))

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Decision Tree

• Binary tree associated with a decision process

 internalnodes: questions with yes/no answer
 external nodes: decisions

• Example: dining decision

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Linked Structure for Binary Trees

• A node is represented by an object storing

 Element
 Parent node

 Left child node

 Right child node

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Codes for Creation of Binary Trees

A binary tree can be created recursively.

The program will work as follow:

o Read a data in x.
o Allocate memory for a new node and
store the address in
pointer p.
o Store the data x in the node p.
o Recursively create the left subtree of p
and make it the left
child of p.
o Recursively create the right subtree of p
and make it the right 28
Codes for Creation of Binary Trees

#include<stdio.h>
typedef struct TreeNode {
struct TreeNode *left;
int data;
struct TreeNode *right;
} TreeNode;

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Codes for Creation of Binary Trees
TreeNode * createBinaryTree( ){
TreeNode *p;
int x;
printf("Enter data(-1 for no data): ");
scanf("%d", &x);
if(x == -1) return NULL;
//create current node
p = (TreeNode*) malloc(sizeof(TreeNode));
p->data = x;
//recursively create left and right subtree
printf("Enter left child of %d: \n", x);
p->left = createBinaryTree ( );
printf("Enter right child of %d: \n",x);
p->right = createBinaryTree ();
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return p; }
Codes for Creation of Binary Tree Traversal
void preorder(TreeNode *t) {
if(t != NULL) {

printf("\n%d", t->data);

preorder(t->left);

preorder(t->right);

}
}

int main() {

TreeNode *root;

root = createBinaryTree ( );

printf("\nThe preorder traversal of tree is: \n");

preorder(root);
} 31
LinkeSd Structure for General Trees

• A node is represented by an object storing

 Element

 Parent node
 Children Container: Sequence of children
nodes

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Thank You

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