Chương này trình bày về đệ quy (recursion). Nội dung trình bày cụ thể gồm có: Đa đệ quy (multiple recursion), Đệ quy chéo (mutual recursion), đệ quy chồng (implicated recursion), nguyên tắc,... Mời các bạn cùng tham khảo để biết thêm chi tiết.
Trang 1< <
! @! > ?ZS AS "S B
hh ! A / 7 Z~"S •
&@
] ?" n B @ @ ?"B
@"7@
! n ?" m Bh
] ?A≤Z~!•B @ @ ? ! @! > ?ZS AS "S !BB
@"7@ @ ? ! @! > ?ZS AS !m<S BB
@ / ]
@ / ]
@ /
' ( 4
Trang 2< 2
≥
×
=
&@
] ? nCB @ !š! n <
@"7@ !š! n A • !š! ?AS L<B
@ / ]
@ /
< ;
' (
.+ /0o
\Cn <S \<n <
\ n \ L<m \L
,
] ] & ? B
&@
] ?? nCB ? n<BB @ ] & n <
@"7@ ] & n ] &? L<B m ] &? L B
@ / ]
@ /
Trang 3< E
= "t
.+ /0
D M 7r h"{
,
−
=
=
−
=
=
&@
] ? nCB @ @'@ n @
@"7@ @'@ n //? L<B
@ / ]
@ /
&@
] ? nCB @ // n ] "7@
@"7@ // n @'@ ? L<B
@ / ]
@ /
< N
' (
$ € ? !>" @/ @ 7 B
.+ /0
=
>
−
= +
=
&@
] ?!nCB @ Z @ ! n m<
@"7@ ] ? nCB Z @ ! n Z @ ! ?!L<S<B
@"7@ Z @ ! n Z @ ! ?!L<S Z @ ! ?!S L<BB
@ / ]
@ / ]
@ /
Trang 4< 5
# :
!= 7r l> ! ( > > A D z ‰ l>
!= 1 1 g != ‰ l> > 6 > Š
& %
T ! > ( l D M # ! P /j "
3 ( > >
' (
.+ /0 <
] / ' 7 ? S &B hh ( 7w •CS &•C
&@
] ? ≥&B @
] ? n&B / ' 7 n @
@"7@ / ' 7 n ] "7@
@ / ]
@"7@ / ' 7 n/ ' 7 ? S &L B
@ / ]
@ /
Trang 5< G
.+ /0
,
&@
] ? nC n<B @ 7 7@ n <
@"7@
] ? ! / n CB 7 7@ n 7 7@? h B
@"7@ 7 7@ n 7 7@?2• m<B
@ / ]
@ / ]
@ /
' (
" X x
.
&= /j " X
V
9 S % e ) /g # &= /j " X [ S % l "
Trang 67
> ! F
&@
@7 " n @ ! @?FB
y "@ ? @7 " n @B
y ?< ! @B
@ /y "@
@ /
' (
? B hh ≥C
&@
] ? nCB @ /
@"7@
? L<B
> ? B hh
@ / ]
@ /
3? B hh ≥C
&@
] ? nCB @ /
@"7@
> ? B hh 3? L<B
@ / ]
@ /
Trang 7p
M
<2;
' (
9 ( $ % l & '4 L< M
# :
1 L< M p e g Z 7
1 M "4 H g Z 7
1 L< M p e g 7
,
T ? S ZS S B
&@
] ? n<B @ 1 M "4 g Z 7
@"7@ T ? L<S ZS S B
1 M "4 g Z 7
T ? L<S S ZS B
@ / ]
@ /
Trang 8+ 7r "J 1 M
− + +
−
=
=
? B n L<
<2N
' (
> + $ ?<B
s
]?AS S •B
s
]? S &S B
s
@ /
<Q I " j (
Q v A%> "
! 7rS D u $ 'PS s
;Q IH g v A%>
EQ 8
NQ v A%> 6 % $ ( ( 'P 5Q F 'P 'D + !
`Q IH % $ ( g v A%>
GQ ) 0 € " j (
Trang 9> + $ ? B
.+ /0
] "? B
&@
] " n <
] n
] " n ] " •
@ /]
@ /
] "? B
&@
] ? n <B @ ] " n <
@"7@
] " n • ] "? L<B
@ / ]
@ / L< > Y> *
> Y>
> Y> ?'Π"q>B
L< > Y> v < ?'Π"q>B
> Y> 7 7
L< > Y> *
> Y>
L< > Y> g ? + L<B
> Y> 7 7
G >
> + $ H "4 / " !
[w /0 'Π"q>
&H „ Aw "† 7 " $
Aw "† 7 " $
Trang 10)?aB
&@
X )?α?aBB
@"7@
@ / ]
@ /
o P > 0 = a
X o Aw "† e&( K ,
α?aB o & 1 / 78 1 c ! 7r
o Aw "† /g
)ž?aB
&@
y "@ / X
a nα?aB
@ /y "@
@ /
&7@ ?‡S ZS "S B
&@
] ?"≤ B @
! n ?"m Bh
] ?‡ n Z~!•B @ &7@ n !
@"7@ ] ?‡ – Z~!•B @ &7@ n &7@ ?‡S ZS "S !L<B
@"7@ &7@ n &7@ ?‡S ZS !m<S B
@ / ]
@ / ]
@"7@
&7@ n C
@ / ]
@ /
Trang 11&7@ ž?‡S ZB
&@
" n <
n
y "@ ?"≤ B /
! n ?"m Bh ] ?‡ n Z~!•B @ &7@ ž n !‚ @
@"7@ ] ?‡ – Z~!•B @ n !L<
@"7@ " n !m<
@ / ]
@ / ]
@ /y "@
&7@ ž n C
@ /
1 " 8 %> K " $
'4 v A%>
@ @ 7@!>
> 7
> >
>
Trang 12<
F?aB
&@
] ?aB @
?aB F?α?aBB U?aB
@"7@
?aB
@ / ]
@ /
Fž?aB
&@
@ @?[B
y "@ ?aB /
?aB
> 7 ?[S aB
a nα?aB
@ /y "@
?aB
y "@ 7@!> ?[B /
a n >?[B U?aB
> >?[B
@ /y "@
@ /
<
W
3 F?aCB
?aCB 9
?aCB
3 F?α?aCBB
?α?aCBB 9
?α?aCBB
3 F?α?α?aCBBB
?α?α?aCBBB 7
?α?α?aCBBB U?α?aCBB
U?α?α?aCBB
3 Fž?aCB x
Trang 13<
.+ /0
? B hh ≥C
&@
] ? nCB @ /
@"7@
? L<B
> ? B hh
@ / ]
@ /
ž? B hh ≥C
&@
@ @?[B ] ? nCB @ /
@"7@
y "@ ? •CB /
> 7 ?[S B
n L<
@ /y "@
y "@ ? 7@!> ?[BB /
n >?[B
> ? B hh
> >?[B
@ /y "@
@ / ]
@ /
<;N
F?aB
&@
] ?aB @
?aB F?α?aBB
U?aB
@"7@
?aB
@ / ]
@ /
Fž?aB
&@
@ @?[B
> 7 ?[S ? @y ""S aBB
y "@ 7@!> ?[B /
?7 @S B n >?[B
> >?[B ] ?7 @ n @y ""B @
a n ] ?aB @
?aB
> 7 ?[S ?@ /S aBB
> 7 ?[S ? @y ""Sα?aBBB
@"7@ ?aB
@ / ]
@ / ] ] ?7 @ n @ /B @
a n U?aB
@ / ]
@ /y "@
@ /
Trang 14W
3 F?aCB
?aCB 9
?aCB
3 F?α?aCBB
?α?aCBB 9
?α?aCBB
3 F?α?α?aCBBB
?α?α?aCBBB 7
?α?α?aCBBB U?α?aCBB
U?α?α?aCBB
3 Fž?aCB x
<;`
.+ /0
? B hh ≥C
&@
] ? nCB @ /
@"7@
? L<B
> ? B hh
@ / ]
@ /
ž? B
&@
@ @?[B
> 7 ?[S ? @y ""S BB
y "@ 7@!> ?[B /
?7 @S B n >?[B
> >?[B ] ?7 @ n @y ""B @ ] ? •CB @
> 7 ?[S ?@ /S BB
> 7 ?[S ? @y ""S L<BB
@"7@ /
@ / ]
@ / ] ] ?7 @ n @ /B @
> ? B
@ / ]
@ /y "@
@ /
Trang 15e
,> ?<B
<
D M /i 7r \ &
\ &Cn <S \ &<n <
\ & n \ & L<m \ &L
Ti 8
<Q ‡* /8 , $ + \ &? B
Q 6 ! k = > 6 > ?&f 7r > Y> = B K ,
" Ω ? h B
2Q ‡* /8 , + q> ?\ &? BS \ &? L<BB '4 • C
+ \ &? B
EQ = > 6 > ?&f 7r > Y> = B K , #