Tree Data Structures

Topics to be discussed….

•Trees Data Structures•Trees•Binary Search Trees•Tree traversal•Types of Binary Trees•Threaded binary trees•Applications

Trees Data Structures

Tree◦ Nodes◦ Each node can have 0 or more children◦ A node can have at most one parent

Binary tree◦ Tree with 0–2 children per node

Tree Binary Tree back

Trees Terminology

◦ Root no parent◦ Leaf no child◦ Interior non-leaf◦ Height distance from root to leaf

Root node

Leaf nodes

Interior nodes Height

Level and Depth

K L

E F

B

G

C

M

H I J

D

ALevel

1

2

3

4

node (13)degree of a nodeleaf (terminal)Non terminalparentchildrensiblingdegree of a tree (3)ancestorlevel of a nodeheight of a tree (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

Arithmetic Expression Using BT+

*

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

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Binary Search Trees

Key property◦ Value at node

Smaller values in left subtree Larger values in right subtree

◦ Example X > Y X < Z

Y

X

Z

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Tree traversal

Two types of tree traversal are;

1.Recursive

2.Non recursive

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Preorder: node, left, right

Inorder: left, node, right

Postorder: left, right, node

Recursive traversal

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In this stack is used to implement the non recursive traversal.

All nodes first put into the stack and then each node is processed according to traversal.

Non recursive traversal

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Binary Search Trees

Binary search trees

Not a binary search tree

5

10

30

2 25 45

5

10

45

2 25 30

5

10

30

2

25

45

Example Binary Searches Find (root, 25 )

5

10

30

2 25 45

5

10

30

2

25

45

10 < 25, right

30 > 25, left

25 = 25, found

5 < 25, right

45 > 25, left

30 > 25, left

10 < 25, right

25 = 25, found

Types of Binary Trees Degenerate – only one child Complete – always two children Balanced – “mostly” two children

more formal definitions exist, above are intuitive ideas

Degenerate binary tree

Balanced binary tree

Complete binary treeback

Threaded Binary Trees

Two many null pointers in current representationof binary trees n: number of nodes number of non-null links: n-1 total links: 2n null links: 2n-(n-1)=n+1

Replace these null pointers with some useful “threads”.

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Pre-order traversal while duplicating nodes and values can make a complete duplicate of a binary tree. It can also be used to make a prefix expression (Polish notation) from expression trees: traverse the expression tree pre-orderly.

In-order traversal is very commonly used on binary search trees because it returns values from the underlying set in order, according to the comparator that set up the binary search tree (hence the name).

Post-order traversal while deleting or freeing nodes and values can delete or free an entire binary tree. It can also generate a postfix representation of a binary tree.

Applications

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