tree matematika diskrit
TRANSCRIPT
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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!$"* degree 14
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Pembuktian Teorema 2.1.1.2
(nductive step
M*%"+#"$ 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!$"* n ; 1 edge4
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Pembuktian Teorema 2.1.1.3
34 A#"$ )*,#&*#"$ *#" 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!$"* n ; 1 edge4
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Pembuktian Teorema 2.1.1.4
>4 A#"$ )*,#&*#"$ *#" 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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