• Tree Data Structures
Objectives

At the end of this class, you should:

Know what a tree data structure is

Be able to explain fundamental tree terminologies

Know how to traverse a tree

pre-order

In-order

post-order
Objectives

At the end of this class, you should:

Know what a binary tree is

Full/Strict

Complete

Perfect
Objectives

At the end of this class, you should:

Understand Binary Search Trees

How to retrieve from them

How to insert into them

How to delete from them
Intro
A
B
D
C
E
H
G
I
J
Definition

0

1
2
4
7
5
8


Trees are non-linear

Trees represent relationships
6
9
Each node may have zero or more
successors
Each node has exactly one predecessor
except the starting node.
Trees are hierarchical

3
A tree is a collection of nodes.
Tree Terminologies

0
Root:
-
The starting node of a tree, i.e. the
node without any predecessor
Tree Terminologies

0
Parent:
-
1
2

Child:
-
3
4
5
6

8
9
Successor of a node
Siblings:
-
7
Predecessor of a node
Nodes with the same parent
Tree Terminologies

3
5
7
8
9
Leaf:
-
A node without any child
-
Shows the termination of a tree
branch
Tree Terminologies

Branch:
-
The link from one node to another
-
Also called an Edge
Tree Terminologies

Subtree:
-
The subtree of a node is a tree whose
root node is a child of that node
-
Left subtree of the root node
Tree Terminologies

Subtree:
-
The subtree of a node is a tree whose
root node is a child of that node
-
Right subtree of the root node
Tree Terminologies

Depth:
-
Measure of the distance of a node
from the root.
-
Depth 0
Tree Terminologies

Depth:
-
Measure of the distance of a node
from the root.
-
Depth 1
Tree Terminologies

Depth:
-
Measure of the distance of a node
from the root.
-
Depth 2
Tree Terminologies

Depth:
-
Measure of the distance of a node
from the root.
-
Depth 3
Tree Terminologies

Depth:
-
Measure of the distance of a node
from the root.
-
Also called the path-length
-
Depth 3
Tree Terminologies


Depth:
-
Measure of the distance of a node
from the root.
-
Also called the path-length
-
Depth 3
Nodes at the same depth form a level
Tree Terminologies

Height:
-
The maximum distance of any leaf
node from the root node
-
Height = 3
Tree Examples
1
1
2
2
1
3
3
1
4
5
2
5
3
4
2
Binary Tree


A tree in which no node can have
more than two children
Each node either

has at most two non-empty subtrees

or is a leaf node.
Full Binary Tree

A binary three in which each node
has two or zero child nodes.

There is no node with only one child
Complete Binary Tree

A binary tree in which every level,
except possibly the deepest, is
completely filled.

At the height of the tree (the deepest
level), all nodes (leaves) are as far
left as possible
1 2

Where would the next node go?
3
4 5
6
Complete Binary Tree

A binary tree in which every level,
except possibly the deepest, is
completely filled.

At the height of the tree (the deepest
level), all nodes (leaves) are as far
left as possible
1

Where would the next node go?
Perfect Binary Tree



A binary tree with all leaf nodes at
the same depth
All internal nodes have exactly two
children
Has the max no of nodes for a given
height.

(2n+1 – 1) nodes

n = height of the tree
Binary Tree Traversal

Traversal:


Visiting the nodes of a tree
Four traditional types grouped in two

Deapth First

Pre-order

In-order

Post-order

Breadth First

Level-order
Pre-Order Traversal

Visit a node (N)
Pre-Order Traversal

Visit a node (N)

Visit its left subtree (L)
Pre-Order Traversal

Visit a node (N)

Visit its left subtree (L)

Visit its right subtree (R)
Pre-Order Traversal

Visit a node (N)

Visit its left subtree (L)

Visit its right subtree (R)

Preoder ==> NLR

Node – Left – Right
Pre-Order Traversal

Example 1.
A
B

C
Solution

Condition: NLR

Answer:
D
E
H
F
I
G
J
Pre-Order Traversal

Example 1.
A
B

C
Solution

Condition: NLR

Answer:

A,
D
E
H
F
I
G
J
Pre-Order Traversal

Example 1.
A
B

C
Solution

Condition: NLR

Answer:

A, B,
D
E
H
F
I
G
J
Pre-Order Traversal

Example 1.
A
B

C
Solution

Condition: NLR

Answer:

A, B, D,
D
E
H
F
I
G
J
Pre-Order Traversal

Example 1.
A
B

C
Solution

Condition: NLR

Answer:

A, B, D, E,
D
E
H
F
I
G
J
Pre-Order Traversal

Example 1.
A
B

C
Solution

Condition: NLR

Answer:

A, B, D, E, H,
D
E
H
F
I
G
J
Pre-Order Traversal

Example 1.
A
B

C
Solution

Condition: NLR

Answer:

A, B, D, E, H, I,
D
E
H
F
I
G
J
Pre-Order Traversal

Example 1.
A
B

C
Solution

Condition: NLR

Answer:

A, B, D, E, H, I, C,
D
E
H
F
I
G
J
Pre-Order Traversal

Example 1.
A
B

C
Solution

Condition: NLR

Answer:

A, B, D, E, H, I, C, F,
D
E
H
F
I
G
J
Pre-Order Traversal

Example 1.
A
B

C
Solution

Condition: NLR

Answer:

A, B, D, E, H, I, C, F, G,
D
E
H
F
I
G
J
Pre-Order Traversal

Example 1.
A
B

C
Solution

Condition: NLR

Answer:

A, B, D, E, H, I, C, F, G, J
D
E
H
F
I
G
J
Pre-Order Traversal

Example 2.
3
7

9
Solution

Condition: NLR

Answer:
8
4
2
5
Pre-Order Traversal

Example 2.
3
7

9
Solution

Condition: NLR

Answer:

3,
8
4
2
5
Pre-Order Traversal

Example 2.
3
7

9
Solution

Condition: NLR

Answer:

3, 7,
8
4
2
5
Pre-Order Traversal

Example 2.
3
7

9
Solution

Condition: NLR

Answer:

3, 7, 8,
8
4
2
5
Pre-Order Traversal

Example 2.
3
7

9
Solution

Condition: NLR

Answer:

3, 7, 8, 4,
8
4
2
5
Pre-Order Traversal

Example 2.
3
7

9
Solution

Condition: NLR

Answer:

3, 7, 8, 4, 9,
8
4
2
5
Pre-Order Traversal

Example 2.
3
7

9
Solution

Condition: NLR

Answer:

3, 7, 8, 4, 9, 2,
8
4
2
5
Pre-Order Traversal

Example 2.
3
7

9
Solution

Condition: NLR

Answer:

3, 7, 8, 4, 9, 2, 5
8
4
2
5
In-order Traversal

Visit the left subtree (L) of a node
In-order Traversal

Visit the left subtree (L) of a node

Visit the node (N)
In-order Traversal

Visit the left subtree (L) of a node

Visit the node (N)

Visit its right subtree (R)
In-order Traversal

Visit the left subtree (L) of a node

Visit the node (N)

Visit its right subtree (R)

Inoder ==> LNR

Left – Node – Right
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:
D
E
F
G

H
I
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E, I,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E, I, A,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E, I, A,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E, I, A,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E, I, A, F,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E, I, A, F, C,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E, I, A, F, C,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E, I, A, F, C,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E, I, A, F, C, J,
D
E
H
F
I
G
J
In-order Traversal

Example 1.
A
B

C
Solution

Condition: LNR

Answer:

D, B, H, E, I, A, F, C, J, G
D
E
H
F
I
G
J
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:

8,
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:

8, 7,
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:

8, 7,
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:

8, 7, 4,
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:

8, 7, 4, 3,
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:

8, 7, 4, 3,
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:

8, 7, 4, 3,
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:

8, 7, 4, 3, 2,
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Order: L-M-R

Answer:

8, 7, 4, 3, 2, 9,
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Order: L-M-R

Answer:

8, 7, 4, 3, 2, 9,
8
4
2
5
In-order Traversal

Example 2.
3
7

9
Solution

Condition: LNR

Answer:

8, 7, 4, 3, 2, 9, 5
8
4
2
5
Post-order Traversal

Visit the right subtree (L) of a node
Post-order Traversal

Visit the left subtree (L) of a node

Visit its right subtree (R)
Post-order Traversal

Visit the left subtree (L) of a node

Visit its right subtree (R)

Visit the node (M)
Post-order Traversal

Visit the left subtree (L) of a node

Visit its right subtree (R)

Visit the node (M)

Inoder ==> LRN

Left – Right – Node
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H, I,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condtion: LRN

Answer:

D, H, I, E,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H, I, E, B,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H, I, E, B,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H, I, E, B,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H, I, E, B, F,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H, I, E, B, F,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H, I, E, B, F,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H, I, E, B, F, J,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H, I, E, B, F, J, G,
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H, I, E, B, F, J, G, C
D
E
H
F
I
G
J
Post-order Traversal

Example 1.
A
B

C
Solution

Condition: LRN

Answer:

D, H, I, E, B, F, J, G, C, A
D
E
H
F
I
G
J
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:

8,
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:

8,
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:

8, 4,
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:

8, 4, 7,
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:

8, 4, 7,
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:

8, 4, 7,
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:

8, 4, 7, 2,
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:

8, 4, 7, 2,
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:

8, 4, 7, 2, 5,
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:

8, 4, 7, 2, 5, 9,
8
4
2
5
Post-order Traversal

Example 2.
3
7

9
Solution

Condition: LRN

Answer:

8, 4, 7, 2, 5, 9, 3
8
4
2
5
Binary Search Tree (BST)

10
3
2
16
7
6
12
9
22
17
Each node is either empty or has up
to two subtrees
Binary Search Tree (BST)

10
3
2
16
7
6

12
9
22
17
Each node is either empty or has up
to two subtrees
Value in each node > value in all of
its left subtree nodes
Binary Search Tree (BST)

10
3
2
16
7
6

12
9
22
17

Each node is either empty or has up
to two subtrees
Value in each node > value in all of
its left subtree nodes
Value in each node < value in all of
its right subtree nodes
Binary Search Tree (BST)

10
3
2
16
7

12
22


6
9
17
Each node is either empty or has up
to two subtrees
Value in each node > value in all of
its left subtree nodes
Value in each node < value in all of
its right subtree nodes
Left-most node == smallest value
Binary Search Tree (BST)

10
3
16


2
7
6
12
9
22
17
Each node is either empty or has up to
two subtrees
Value in each node > value in all of its
left subtree nodes
Value in each node < value in all of its
right subtree nodes

Left-most node == smallest value

Right-most node == largest value
BST Insertion




Start from the root node
Move to the left child if the value is less, otherwise go to the
right child.
Repeat the comparision until a leaf node is reached
Insert node as a left child if value is less than that of the current
leaf, otherwise insert as a right child.
BST Insertion

Example 1

Before

<10, move left
8
10
10
3
2
7
6
8
16
12
9
22
17
3
2
16
7
6
12
9
22
17
BST Insertion

Example 1

Before

>3, move right
8
10
10
3
3
16
16
8
2
7
6
12
9
22
17
2
7
6
12
9
22
17
BST Insertion

Example 1

Before

>7, move right
8
10
10
3
2
3
16
7
12
22
2
16
7
12
22
8
6
9
17
6
9
17
BST Insertion

Example 1

Before

<9, move left
8
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
8
22
17
BST Insertion

Example 1

Before

9 is leaf, insert
8
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
8
22
17
BST Insertion

Example 2

Before

After
23
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
23
BST Deletion

Traverse the tree, find the node and delete it.


BST properties must be preserved!!
Three cases:

Deleting leaf node

Deleting a node with one child

Deleting a node with two children
BST Deletion

Deleting a leaf node

Traverse to the node’s parent

Remove link to the node
BST Deletion

Example 1

Before

After
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
22
17
BST Deletion

Example 2

Before

After
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
22
9
17
BST Deletion

Node with one child

Traverse to the node

Link child of the node to the parent of the node

Remove the link between the node and its parent

Remove the link between the node and its child
BST Deletion

Example 1

Before

Link parent to child
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 1

Before

Unlik from parent
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 1

Before

Unlink from child
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 1

Before

Complete
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
17
BST Deletion

Node with two children

Find the node to be deleted


Let’s call it node x
Find the rightmost node on its left subtree

Let’s call this node z

The leftmost node on the right subtree can be used instead

Replace the node x data with node z data

Delete node z
BST Deletion

Example 1: using the rightmost node in the left subtree

Before

Locate node z
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 1: using the rightmost node in the left subtree

Before

Locate node z
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 1: using the rightmost node in the left subtree

Before

Locate node z
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 1: using the rightmost node in the left subtree

Before

Locate node z
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 1: using the rightmost node in the left subtree

Before

Substitute node z data
9
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 1: using the rightmost node in the left subtree

Before

Delete node z
9
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 1: Using the rightmost node in the left subtree

Before

Complete
9
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
22
17
BST Deletion

Example 2: using the leftmost node in the right subtree

Before

Locate node z
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 2: using the leftmost node in the right subtree

Before

Locate node z
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 2: using the leftmost node in the right subtree

Before

Locate node z
10
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 2: using the leftmost node in the right subtree

Before

Substitute node z data
12
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 2: using the leftmost node in the right subtree

Before

Delete node z
12
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
12
9
22
17
BST Deletion

Example 2: using the leftmost node in the right subtree

Before

Complete
12
10
3
2
7
6
3
16
12
9
22
17
2
16
7
6
22
9
17
Closing

Trees

are heirachichal data structures

are collection of nodes.

show the relationship between their nodes

are awesome