1

Trees

• What is a Tree?

• Tree terminology

• Why trees?

• What is a general tree?

• Implementing trees

• Binary trees

• Binary tree implementation

• Application of Binary trees

2

What is a Tree?

• A tree, is a finite set of nodes together with a finite set of directed edges that define parent-child relationships. Each directed edge connects a parent to its child. Example:

Nodes={A,B,C,D,E,f,G,H}

Edges={(A,B),(A,E),(B,F),(B,G),(B,H),

(E,C),(E,D)}

• A directed path from node m1 to node mk is a list of nodes m1, m2, . . . , mk such that each is the parent of the next node in the list. The length of such a path is k - 1.

• Example: A, E, C is a directed path of length 2.

A

BE

CD F H G

3

What is a Tree? (contd.)

• A tree satisfies the following properties:

1. It has one designated node, called the root, that has no parent.

2. Every node, except the root, has exactly one parent.

3. A node may have zero or more children.

4. There is a unique directed path from the root to each node.

5

2

4 1 6

3

5

2

4 1 6

3

5

2

4

1

6

3

tree Not a tree Not a tree

4

Tree Terminology

• Ordered tree: A tree in which the children of each node are linearly ordered (usually from left to right).

• Ancestor of a node v: Any node, including v itself, on the path from the root to the node.

• Proper ancestor of a node v: Any node, excluding v, on the path from the root to the node.

A

CB

ED F G

Ancestors of Gproper ancestors of E

An Ordered Tree

5

Tree Terminology (Contd.)

• Descendant of a node v: Any node, including v itself, on any path from the node to a leaf node (i.e., a node with no children).

• Proper descendant of a node v: Any node, excluding v, on any path from the node to a leaf node.

• Subtree of a node v: A tree rooted at a child of v.

Descendants of a node C

A

CB

ED F G

Proper descendants of node B

A

CB

ED F G

subtrees of node A

6

Tree Terminology (Contd.)

AAB

D

H

CE F G

JI

proper ancestors of node H

proper descendants of node C

subtrees of A

AA

B

D

H

C

EF

G

JI

parent of node D

child of node D

grandfather of nodes I,J

grandchildren of node C

7

Tree Terminology (Contd.)

• Each of node D and B

has degree 1.• Each of node A and E

has degree 2.

• Leaf: A node with degree 0.• Internal or interior node: a node with degree greater than 0.• Siblings: Nodes that have the same parent.• Size: The number of nodes in a tree.

AA

B

D

H

C

EF

G

JI

• node C has degree 3.

• Each of node F,G,H,I,J

has degree 0.

An Ordered Tree with size of 10

Siblings of E

Siblings of A

8

Tree Terminology (Contd.)

• Level (or depth) of a node v: The length of the path from the root to v.• Height of a node v: The length of the longest path from v to a leaf

node.– Height of a node v: The length of the longest path from v to a leaf

node.– The height of a tree is the height of its root mode.– By definition the height of an empty tree is -1.

AA

B

D

H

C

E F G

JI

k

Level 0

Level 1

Level 2

Level 3

Level 4

• The height of the tree is 4.

• The height of node C is 3.

9

Why Trees?

• Trees are very important data structures in computing.• They are suitable for:

– Hierarchical structure representation, e.g.,• File directory.• Organizational structure of an institution.• Class inheritance tree.

– Problem representation, e.g.,• Expression tree.• Game tree.• Decision tree.

– Efficient algorithmic solutions, e.g.,• Search trees.• Efficient priority queues via heaps.

10

Implementing Trees

• The various classes of trees are viewed as classes of containers as shown below:

MyComparable

Container

Tree

AbstractObject

AbstractContainer

AbstractTree

GeneralTree

NaryTree

BinaryTree

11

Implementing Trees (Contd.)

• The tree interface is defined as:

1 public interface Tree extends Container

2 {

3 Object getKey( ) ;

4 Tree getSubtree(int i) ;

5 boolean isEmpty( ) ;

6 boolean isLeaf( ) ;

7 int getDegree( ) ;

8 int getHeight( ) ;

9 void depthFirstTraversal(PrePostVisitor visitor) ;

10 void breadthFirstTraversal(Visitor visitor) ;

11 }

12

Implementing Trees (Contd.)

• The AbstractTree class skeleton is:

1 public abstract class AbstractTree extends AbstractContainer implements Tree

2 {

3 public void depthFirstTraversal(PrePostVisitor visitor){

4 // . . .

5 }

6 public void breadthFirstTraversal(Visitor visitor){

7 // . . .

8 }

9 public void accept(Visitor visitor){

10 // . . .

11 }

12 public Enumeration getEnumeration( ){

13 // . . .

14 }

15 // . . .

16 }

13

General Trees

• A general tree is an ordered tree whose set of nodes cannot be empty.

• In a general tree, there is no limit to the number of children that a node can have.

• Representing a general tree by linked lists:– Each node has a linked list of the subtrees of that node.

1 public class GeneralTree extends AbstractTree

2 {

3 protected Object key ;

4 protected int degree ;

5 protected LinkedList list ;

6 // . . .

7 }

14

General Trees (Contd.)

• An example of a general tree:

D

E

F H

G

J

K L

M

K

15

N-ary Trees

• An N-ary tree is an ordered tree that is either:1. Empty, or2. It consists of a root node and at most N non-empty N-ary

subtrees.• It follows that the degree of each node in an N-ary tree is at most N.• Example of N-ary trees:

5

72

59 2-ary (binary) tree

B

F

J

DD

C G AEB3-ary (tertiary)tree

16

N-ary Trees (Contd.)

1 public class NaryTree extends AbstractTree{2 protected Object key ;3 protected int degree ;4 protected NaryTree[ ] subtree ;5 public NaryTree(int degree){6 key = null ; this.degree = degree ;8 subtree = null ;9 }10 public NaryTree(int degree, Object key){11 this.key = key ;12 this.degree = degree ;13 subtree = new NaryTree[degree] ;14 for(int i = 0; i < degree; i++) subtree[i] = new NaryTree(degree);15 }16 // . . .17 }

17

Binary Trees

• A binary tree is an N-ary tree for which N = 2.• Thus, a binary tree is either:

1. An empty tree, or

2. A tree consisting of a root node and at most two non-empty binary subtrees.

5

72

59

Example:

18

Binary Trees (Contd.)

• A full binary tree is either an empty binary tree or a binary tree in which each level k, k > 0, has 2k nodes.

• A complete binary tree is either an empty binary tree or a binary tree in which:

1. Each level k, k > 0, other than the last level contains the maximum number of nodes for that level, that is 2k.

2. The last level may or may not contain the maximum number of nodes.

3. If a slot with a missing node is encountered when scanning the last level in a left to right direction, then all remaining slots in the level must be empty.

• Thus, every full binary tree is a complete binary tree, but the opposite is not true.

19

Binary Trees (Contd.)

• Example showing the growth of a complete binary tree:

20

Binary Trees Implementation

• A binary tree can be represented using a linked list.

1 public class BinaryTree extends AbstractTree{ 2 protected Object key ; 3 protected BinaryTree left , 4 protected BinaryTree right ; 5 public BinaryTree(Object key, BinaryTree left, BinaryTree right){ 6 this.key = key ; 7 this.left = left ; this.right = right ; 8 } 9 public BinaryTree( ) 10 { this(null, null, null) ;} 11 12 public BinaryTree(Object key){ 13 this(key, new BinaryTree( ), new BinaryTree( )); } 14 // . . . 15 }

Example: A binary tree representing a + (b - c) * d

leftkeyright

+

a *

d-

b c

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Binary Trees Implementation (Contd.)

• Some methods of BinaryTree class: 1 public boolean isEmpty( ) 2 { return key == null ; } 3 public boolean isLeaf( ) 4 { return ! isEmpty( ) && left.isEmpty( ) && right.isEmpty( ) ; } 5 public Object getKey( ){ 6 if(isEmpty( )) 7 throw new InvalidOperationException( ) ; 8 else 9 return key ;} 10 11 public int getHeight( ){ 12 if(isEmpty( )) 13 return -1 ; 14 else 15 return 1 + Math.max(left.getHeight( ), right.getHeight( )) ; 16 } 17 public void attachKey(Object obj){ 18 if(! isEmpty( )) 19 throw new InvalidOperationException( ) ; 20 else{ 21 key = obj ; 22 left = new BinaryTree( ) ; 23 right = new BinaryTree( ) ; 24 } 25 }

22

Binary Trees Implementation (Contd.)

26 public Object detachKey( ) 27 { 28 if(! isLeaf( )) 29 throw new InvalidOperationException( ) ; 30 else 31 { 32 Object obj = key ; 33 key = null ; 34 left = null ; 35 right = null ; 36 return obj ; 37 } 38 } 39 public BinaryTree getLeft( ) 40 { 41 if(isEmpty( )) 42 throw new InvalidOperationException( ) ; 43 else 44 return left ; 45 } 46 public BinaryTree getRight( ) 47 { 48 if(isEmpty( )) 49 throw new InvalidOperationException( ) ; 50 else 51 return right ; 52 }

23

Application of Binary Trees

• Binary trees have many important uses. Two examples are:

1. Binary decision trees.• Internal nodes are conditions. Leaf nodes denote decisions.

• Expression Tree

Condition1

Condition2

Condition3

decision1

decision2

decision3 decision4

false

false

false

True

True

True

+

a *

d-

b c