Data Structures & Algorithm Analysis

Muhammad Hussain MughalMuhammad.hussain@nu.edu.pk

Trees. Binary Trees.Reading: Chap.4 (4.1-4.2) Weiss

Books

Books

Books

More Trees Examples

Unix / Windows file structure

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.

Level and Depth

K L

E F

B

G

C

M

H I J

D

A

Level

1

2

3

4

node (13)degree of a nodeleaf (terminal)nonterminalparentchildrensiblingdegree 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

Terminology

The degree of a node is the number of subtreesof the node

The degree of A is 3; the degree of C is 1.

The node with degree 0 is a leaf or terminal node.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.

Tree Properties

A

B C

D

G

E F

IH

Property ValueNumber of nodesHeightRoot NodeLeavesInterior nodesNumber of levelsAncestors of HDescendants of BSiblings of ERight subtree

Representation of Trees

List Representation( A ( B ( E ( K, L ), F ), C ( G ), D ( H ( M ), I, J ) ) )The root comes first, followed by a list of sub-trees

data link 1 link 2 ... link n

How many link fields are needed in such a representation?

A Tree Node

Every tree node:object – useful informationchildren – pointers to its children nodes

O

O O

O

O

Left Child - Right Sibling

A

B C D

E F G H I J

K L M

data

left child right sibling

Tree ADT

Objects: any type of objects can be stored in a treeMethods:accessor methods

root() – return the root of the treeparent(p) – return the parent of a nodechildren(p) – returns the children of a node

query methodssize() – returns the number of nodes in the tree isEmpty() - returns true if the tree is emptyelements() – returns all elementsisRoot(p), isInternal(p), isExternal(p)

Tree Traversal

Two main methods:PreorderPostorder

Recursive definition

PREorder: visit the roottraverse in preorder the children (subtrees)

POSTordertraverse in postorder the children (subtrees)visit the root

Trees

Tree traversals:Inorder traversal – prints the node values in ascending order

1. Traverse the left subtree with an inorder traversal

2. Process the value in the node (i.e., print the node value)

3. Traverse the right subtree with an inorder traversal

Preorder traversal1. Process the value in the node2. Traverse the left subtree with a preorder

traversal3. Traverse the right subtree with a preorder

traversal

Postorder traversal1. Traverse the left subtree with a postorder

traversal2. Traverse the right subtree with a postorder

traversal3. Process the value in the node

Preorder

preorder traversalAlgorithm preOrder(v)

“visit” node vfor each child w of v do

recursively perform preOrder(w)

Postorder

postorder traversalAlgorithm postOrder(v)

for each child w of v do recursively perform postOrder(w)“visit” node v

du (disk usage) command in Unix

Binary Trees

A special class of trees: max degree for each node is 2Recursive definition: A binary tree is a finite set of nodes that is either empty or consists of a root and two disjoint binary trees called the left subtree and the right subtree.Any tree can be transformed into binary tree.

by left child-right sibling representation

Example

J

IM

HL

A

B

C

D

E

F G K

Samples of Trees

A

B

A

BA

B C

GE

I

D

H

F

Complete Binary Tree

Skewed Binary Tree

E

C

D

1

2

3

45

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 nubmer of nodes in a binary tree of depth k is 2k-1, k>=1.

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

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

Sequential Representation

AB--C------D--.E

[1][2][3][4][5][6][7][8][9].[16]

[1][2][3][4][5][6][7][8][9]

ABCDEFGHI

A

B

E

C

D

A

B C

GE

I

D

H

F

(1) waste space(2) insertion/deletion problem

Linked Representationtypedef struct tnode *ptnode;typedef struct tnode { int data; ptnode left, right;};

dataleft

right

data

left right

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

lRr, lrR, Rlr, Rrl, rRl, rlR

Adopt convention that we traverse left before right, only 3 traversals remain

lRr, lrR, Rlrinorder, postorder, preorder

CS 103 26

Illustrations for Traversals

Assume: visiting a node is printing its labelPreorder: 1 3 5 4 6 7 8 9 10 11 12Inorder:4 5 6 3 1 8 7 9 11 10 12Postorder:4 6 5 3 8 11 12 10 9 7 1

1

3

11

98

4 6

5

7

12

10

CS 103 27

Illustrations for Traversals (Contd.)

Assume: visiting a node is printing its dataPreorder: 15 8 2 6 3 711 10 12 14 20 27 22 30Inorder: 2 3 6 7 8 10 1112 14 15 20 22 27 30Postorder: 3 7 6 2 10 1412 11 8 22 30 27 20 15

6

158

2

3 7

11

10

14

12

20

27

22 30

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

Inorder Traversal (recursive version)

void inorder(ptnode ptr)/* inorder tree traversal */{ if (ptr) { inorder(ptr->left); printf(“%d”, ptr->data); indorder(ptr->right); }}

A / B * C * D + E

Preorder Traversal (recursive version)

void preorder(ptnode ptr)/* preorder tree traversal */{ if (ptr) { printf(“%d”, ptr->data); preorder(ptr->left); predorder(ptr->right); }}

+ * * / A B C D E

Postorder Traversal (recursive version)

void postorder(ptnode ptr)/* postorder tree traversal */{ if (ptr) { postorder(ptr->left); postdorder(ptr->right); printf(“%d”, ptr->data); }}

A B / C * D * E +

Level Order Traversal(using queue)

void levelOrder(ptnode ptr)/* level order tree traversal */{ int front = rear = 0; ptnode queue[MAX_QUEUE_SIZE]; if (!ptr) return; /* empty queue */ enqueue(front, &rear, ptr); for (;;) { ptr = dequeue(&front, rear);

if (ptr) { printf(“%d”, ptr->data); if (ptr->left) enqueue(front, &rear, ptr->left); if (ptr->right) enqueue(front, &rear, ptr->right); } else break; }}

+ * E * D / C A B

Application: Evaluation of Expressions

+

*

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