TREE MATEMATIKA DISKRIT

8/19/2019 TREE MATEMATIKA DISKRIT

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EE EE EE

KELOMPOK III

ALBERTUS REVOLIKO S M0113003

BAGUS NAUFAL FAUZI

ISCA YUNITASARI M0113023

RIRIN DWI UTAMITRI INDRAYANA PRASETYA M0113051


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Defnisi

Definisi 2.1.4. Tree mer!"#"$ %"& 'r"( connected )"$ &*)"#

mem*+*#* cycle.

Definisi 2.1.5. Forest mer!"#"$ #m!+"$ )"r* tree

De(*$*%* tree )"$ forest )*"m,*+ )"r* C-"r&r"$)


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Tree Tree B#"$ 'r"( Tree Forest

Defnisi Tree


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Definisi 2.1.6. Term*$"+ .er&e/ )*%e,& '" +e"( mer!"#"$ .er&e/

!")" &ree "$' mem!$"* )e'ree 14

I$&er$"+ .er&e/ ")"+"- .er&e/ !")" &ree "$' mem!$"*)e'ree +e,*- )"r* 1.

Definisi 2.1.7. R&e) &ree ")"+"- %"& &ree )*m"$" &er)"!"& %"+"-

%"& .er&e/ &er&e$& "$' )*")*#"$ %e,"'"* "#"r "$' )*%e

r&4

Definisi 2.1.8. Le.e+ )"r* %e,"- .er&e/ mer!"#"$ m+"- e)'e

)*%e!"$"$' %"& !"&- &$''"+ "$&"r" .er&e/ &er%e,&

)e$'"$ r&

Definisi 2.1.9. 6e*'-& )"r* %"& r&e) &ree ")"+"- m"#%*mm +e.e+

"$' &er)"!"& !")" &ree &er%e,&.

D*"m,*+ )"r* S%%"$"7 &er)"!"& ,e,er"!" &erm*$+'* me$'e$"* &ree4


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Defnisi Tree

r&

+e"(

B

.e


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Defnisi Tree

Definisi 2.1.8. Le.e+ )"r* %e,"- .er&e/ mer!"#"$ m+"- e)'e

)*%e!"$"$' %"& !"&- &$''"+ "$&"r" .er&e/&er%e,& )e$'"$ r&

r&

Le.e+ 1Le.e+ 1

Le.e+ 2

Le.e+ 3

Le.e+ 0


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Defnisi Tree

Definisi 2.1.9. 6e*'-& )"r* %"& r&e) &ree ")"+"- m"#%*mm +e.

"$' &er)"!"& !")" &ree &er%e,&.

r&

Heigh& = 3


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Defnisi TreeDefinisi 2.1.10. Children )"r* . !")" %"& rooted tree ")"+"- %em

vertex "$' adjacent )e$'"$ vertex v. Children mem*+*#

%"& +e.e+ +e,*- &*$''* )"r*!")" vertex v

Definisi 2.1.11. 9*#" vertex w ")"+"- child )"r* vertex v, m"#" vertex

)*%e,& %e,"'"* parent )"r* vertex v.

Definisi 2.1.12. Dua vertex "$' ,er,e)" "$' #e)"$" mer!"#"children )"r* parent "$' %"m" )*%e,& %e,"'"* siblings

Definisi 2.1.13. M*%"+ . )"$ : ")"+"- leaf vertex "$' ,er,e)" !")

rooted tree T, *#" .er&e/ . &er+e&"# !")" path &$''"+

"$&"r" vertex w )"$ root m"#" vertex v mer!"#"$ ancestor )"r* vertex w )"$ vertex w mer!"#"$ descendan

)"r* vertex v.


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Defnisi Tree

Child )"r* .er&e/ ,

)"$ .er&e/ e7 (

")"+"- %"&

siblings

Parent )"r* .er&e/

e )"$ (

Vertex , mer!"#"$

ancestor )"r* .er&e/ ( Vertex ( mer!"#"$

descendant )"r* .er&e/ ,


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Teorema

Teoe!" 2.1.1.

isal T adalah graf dengan n vertex.

Pernyataan beri!ut ini e!uivalen.

". T adalah tree.

#. T connected dan acyclic.

$. T connected dan %e%punyai n ; 1 edge.

&. T acyclic dan %e%punyai n ; 1 edge.


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Pembuktian Teorema 2.1.1.

14 A#"$ )*,#&*#"$ *#" T ")"+"- tree m"#" T connected )"$ acyclic 4

M*%"+ T ")"+"- tree4 Ber)"%"r#"$ Definisi 2.1.4 7 e+"% ,"-:" *#"

")"+"- %"& tree m"#" T connected )"$ acyclic &*)"# mem!$"

cycle4

24 A#"$ )*,#&*#"$ *#" T connected )"$ acyclic m"#" T connected )"

mem!$"* n;1 edge4M*%"+ T connected )"$ acyclic 4

A#"$ )*&$##"$ ,"-:" T mem!$"* n ; 1 e)'e )e$'"$ *$)#%*

m"&em"&*#"


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Pembuktian Teorema 2.1.1.2

'asis step

U$&# n < 17 T &er)*r* )"r* %"& vertex )"$ $+ edge4 9")*7 ,e$"r $&#

(nductive step

D*"%m%*#"$ ,e$"r $&# 'r"( connected )"$ acyclic )e$'"$ ! vertex

%e-*$''" 'r"( &er%e,& mem!$"* ! ;1 edge4

A#"$ )*,#&*#"$ ,e$"r $&# 'r"( connected )"$ acyclic )e$'"$ ! = 1

Am,*+ %e,"r"$' si%ple path P )e$'"$ !"$"$' m"#%*mm4

K"re$" T acyclic 7 P ,#"$ %"& cycle4 O+e- #"re$" *&7 P mem"& %

vertex v "$' mem!$"* degree 14


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(nductive step

M*%"+#"$ T ∗

")"+"- T )e$'"$ vertex v )"$ edge "$' incident &er-")")*-"!%#"$4 D*!er+e- T ∗")"+"- 'r"( "$' connected )"$ acyclic )e$

vertex )"$ ! ; 1 edge

M"#" )*!er+e- T &er)*r* )"r* ! edge4 K"re$" basic step )"$ inductive

&er!e$-*7 )"!"& )*%*m!+#"$ ,"-:" *#" T connected )"$ acyclic m"#

connected )"$ mem!$"* n ; 1 edge4


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34 A#"$ )*,#&*#"$ *#" T connected )"$ mem!$"* n ; 1 edge m"#

acyclic )"$ mem!$"* n ; 1 edge4M*%"+ T connected 7 mem!$"* n ; 1 edge7 )"$ mem"& %e)*#*&$

%"& cycle4

K"re$" )e$'"$ me$'-"!% %e,"- edge )"r* %e,"- cycle &*)"#

mem,"& 'r"( me$")* disconnected 7 m"#" edge )"!"& )*-"!%

%"m!"* mem,e$&# 'r"( T ∗connected )"$ acyclic 4

T ∗mer!"#"$ 'r"( "$' connected )"$ acyclic )e$'"$ n vertex 4

9")* &er,#&* ,"-:" *#" T connected )"$ mem!$"* n ; 1 edge

m"#" T acyclic )"$ mem!$"* n ; 1 edge4


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>4 A#"$ )*,#&*#"$ *#" T acyclic )"$ mem!$"* n ; 1 edge m"#"

")"+"- tree4 Ar&*$"7 -"r% )*!er+*-""$ T ")"+"- %"& 'r"( si%ple )"$

mem!$"* si%ple path &$''"+ )"r* %e,"r"$' vertex #e vertex

+"*$$"4

Gr"( T &*)"# )"!"& mem"& %e,"r"$' loop #"re$" loop ")"+"- c

)"$ T acyclic 4

De$'"$ 8"r" "$' %"m" T &*)"# )"!"& mem"& edge "$' ,er,

e1 )"$ e2 "$' incident )e$'"$ v )"$ w 7

#"re$" "#"$ )*!er+e- cycle v,e1,w,e2,v 4

O+e- #"re$" *& T ")"+"- %"& 'r"( si%ple4


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