fcs 04 trees part01

7/25/2019 FCS 04 Trees Part01

1/22

Foundation of Computing Systems

Lecture 4

Trees: Part I


2/22

31.07.09 IT 60101: Lecture #4 2

Tree: Example

j a n

f e b m a r

a p rm a y j u n

j u l a u g

s e p

c t

! e cn "

#

$

% &

'

( )

* +


3/22

31.07.09 IT 60101: Lecture #4 3

Tree: Definition

, %tree-s a f-n-te set f ne rmre n!es suc tat:

, /- tere -s a spec-ally!es-gnate! n!ecalle! te

rt

, /-- rema-n-ng n!es arepart-t-ne! -nt n/n 0!-sj-nt sets T1 T2 . . . Tnere eac Ti/i 1 2 . . .

n-s a tree5 T1 T2 . . . Tnare calle! sub trees f tert.

T T T

%

&

*

+

)

8 I

: L

1 32

T /A/B/E F/K L C/G D/H I J


4/22

31.07.09 IT 60101: Lecture #4 4

Binary Tree

, % binary tree-s a spec-al frm f atree

, % b-nary tree T-s a f-n-te set fn!es suc tat

, /- T-s empty /calle! empty b-narytree r

, /-- Tcnta-ns a spec-ally !es-gnate!n!e calle! te rtf T an! terema-n-ng n!es f Tfrm t

!-sj-nt b-nary trees T1

an! T2

-care calle! left sub$tree an! te r-gtsub$trees respect-"ely.

%

& *

+ )

8

I

:


5/22

31.07.09 IT 60101: Lecture #4 ;

Full Binary Tree

% f u l l b - n a r y t r e e f e - g t 4

L e v e l N o d e s

1

2 3

4 5 6 7

8 9 1 0 1 1 1 2 1 3

1 4 1 5

0 1

1 2

2 4

3 8


6/22

31.07.09 IT 60101: Lecture #4 6

Complete Binary Tree

L e v e l N o d e s

1

2 3

4 5 6 7

8 9 1 0 1 1 1 2

0 1

1 2

2 4

3 5

% c m p l e t e b - n a r y t r e e f . e - g . t 4


7/22

31.07.09 IT 60101: Lecture #4 7

Skewed Binary Tree


8/22

31.07.09 IT 60101: Lecture #4 -mum number f n!es n le"el l-s 2l

ere l? 0.

, =rperty 2

, @a>-mum number f n!es pss-ble -n a b-nary tree f e-gth-s 2hA 1.

, =rperty 3, @-n-mum number f n!es pss-ble -n a b-nary tree f e-gt

h-s h.

, =rperty 4, r any nn$empty b-nary tree -f n-s te number f n!es

an! e-s te number f e!ges ten n e' 1.


9/22

31.07.09 IT 60101: Lecture #4 9

roperties of Binary Trees

, =rperty ;

, r any nn$empty b-nary tree T -f n0-s te number f leaf

n!es /!egree 0 an! n2-s te number f -nternal n!e

/!egree 2 ten n0 n2' 1.

, =rperty 6

, 8e-gt f a cmplete b-nary tree -t nnumber f n!es -s

, =rperty 7, Ttal number f b-nary tree pss-ble -t nn!es -s

)1(log2 +n

n

n

Cn

2

1

1

+


10/22

31.07.09 IT 60101: Lecture #4 10

!epresentation of Binary Trees

, L-near representat-n

A Bs-ng array

, L-nCe! representat-nA Bs-ng l-nCe! l-st structure


11/22

31.07.09 IT 60101: Lecture #4 11

Linear !epresentation of Binary Trees

1. Te rt n!e -s at lcat-n 1.

2. r any n!e -t -n!e> i 1 D i E n /frsme n:

/a =%F)GT/i

r te n!e en i 1 tere -s n parent.

/b L*8IL+/i 2 (i

If 2 (i n ten ias n left c-l!.

/c F*8IL+/i 2 ( i' 1

If 2 ( i' 1 n ten ias n r-gt c-l!.

2i

'

$ (

% & * #

/ a % b - n a r y t r e e

1 2 3 4 ; 6 7 < 9 1 0 1 1 1 2 1 3 1 4 1 ; 1 6

' $ ( % & * # . . . . . . + )

/ b % r r a y r e p r e s e n t a t - n f t e b - n a r y t r e e

.

+ )


12/22

31.07.09 IT 60101: Lecture #4 12

Linked !epresentation of Binary Trees

+ % T %

F * L *

'

$ (

% & * #

+ )

/ a % b - n a r y t r e e

% ! ! r e s s G ! e c n t e n t

; 0

7 ;

4 0

< 9

6 2

4 6

6 6

;

fcs 04 trees part01

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