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avide ∈ AbinSE

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T : 7

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B C DE FG ? @ A

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I I

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A

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A

J H F>LK M < F@ N F@ A < M O N < F@ H P

;Q < N F@ A R ? JS T = P BE JS T = P I I

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B XG

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W

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A ZY N N[ A H P

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`

M M O M ` I I

Q G Q A R \ O N] U ? @ A

V W

? @ A

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M M O M ` I I

Q G Q A R M F < U ? @ A

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<@

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f(a) = g(a, f(fils gauche(a)), f(fils droit(a)))

7 =P 7 < = :9 7 Q 7 ; 7@ 78

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→

^{@ < M}

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f(fils gauche(a)), nb

f(fils droit(a)))

+ # *) $# Y

A +

g(a, z 1 , z 2 ) = z 1 + z 2

* Y

nb

f(a) = 1

est

arbre

vide(a) nb

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f(fils gauche(a)) + nb

f(fils droit(a))

6 7 < = :9 7 Q 7 ; 7@ 7 8 7 8 9 T: 6

;Q < M N @ ? RG O Q Q A B ^ < N] _ <]

C

V Wb

DE JS T = P V W @ ? RG O Q Q A S @ ? RG O Q Q A = I I

Q @ ? RG O Q Q A U ? @ A

V W @ < B XG

O@

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< M O

b I

G Q A I I

< M O

F

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) +

% $ [ ([

avide ∈

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×

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89 T: 6

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C

D C M? FG Y H Y ? @ H Y ? @ I I

< => Y ? @ B C D C M? FG Y H Y ? @ H Y ? @

J H F

>LK M < F@ N F@ A < M O N < F@

H P

;Q < N F@ A R ? J

TS T = P B C M? J

TS T = P I I

Q N F@ A R ? U Y H Y ? @ H Y ? @ V W Y ? @ B XG

O@

W

J H F

>LK M < F@ A ZY N N[ A H P

;Q < <

O < < B ^ < N] _ <]

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J

TS T = P V W

D R V W G Q _ <]

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Q < O < < U Y ? @ V W Y B XG

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W

;Q < G Q A R \ O N] B ^ < N] _ <]

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TS T = P V W S

D R V W G Q _ <]

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Q G Q A R \ O N] U Y ? @ V W Y ? @ B XG

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W

;Q < G Q A R M F < B ^ < N] _ <]

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TS T = P V W =

D R V W G Q _ <]

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Q G Q A R M F < U Y ? @ V W Y ? @ B XG

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W

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;Q < A < R M? M R B ^ < N] _ <]

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m

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o

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0 b

H $#

noeud(o, a 1 , a ₂ ) ∈ ( ⁰ a, ⁰ b) AbinII

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5

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Y T Y ? P ? @ B

FG Y DE FG Y ? H J

Y T Y ? P ? @ H J

Y T Y ? P ? @ I I

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>LK M < F@ N F@ A < M O N < F@

H P

;Q < N F@ A RG O Q Q J

P B J

P I I

Q N F@ A RG O Q Q U Y V W J

Y T Y ? P ? @ B XG O@ W

;Q < N F@ A R@ F O J N TS T = P BE J N TS T = P I I

Q N F@ A R@ F O U

Y ? H J

Y T Y ? P ? @ H J

Y T Y ? P ? @ V W J

Y T Y ? P ? @ B

5 5

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P V W

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W

;Q < < R@ F O B ^ < N] _ <]

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T

S T = P V W

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Q < R@ F O U J

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;Q < G Q A R \ O N] B ^ < N] _ <]

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TS T = P V W S

D R V W G Q _ <]

`

M M

O M ` I I 5

Q G Q A R \ O N] U J

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Y T Y ? P ? @ B XG O@ W

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T

S T = P V W =

D R V W G Q _ <]

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Q G Q A R M F < U J

Y T Y ? P ? @ V W J

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W

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>LK M < F@ < A < H P

;Q < A < RG O Q Q B ^ < N] _ <]

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I I

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W

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< => F> R M <]

B Q O A D O

Q D F O A D 5 ^

;Q < M?b E J OQ T J P T J P P I I

Q M?b U J @ < T F

> R M <]

P ? @ BE J OQ T T P

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Q M? U J

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A B! ! # $

+# % $

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)

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b

D D

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()

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avide ∈

^{Y M? M}

m ∈

^Y

l ∈

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7 7 =P 7

@ F O J T

T

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5 )

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×

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→

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3 / +

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→

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M

89 T: 6

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C

DE FG Y H Y M? M Q A < I I

< => Y

M? M B C DE FG Y H Y

M? M Q A <

J H F>LK M < F@ N F@ A < M O N < F@ H P

;Q < N F@ A R M? M R B C I I

Q N F@ A R M? M R U Y M? M B C

;Q < N

F@

A R@ F O J

T

Q P BE J

T

Q P I I

Q N F@ A R@ F O U Y H Y M? M Q A < V W Y M? M B XG O@ W

J H F>LK M < F@ < A < H P

;Q < A < R M? M R B ^ < N] _ <]

C V W < M O

D R V W G Q A I I

Q A < R M? M R U Y M? M V W ? F FQ B XG

O@

W

J H F

>LK M < F@ A ZY N N[ A H P

;Q < M N @ B ^ < N] _ <]

E J

T

Q P V W

D R V W G Q _ <]

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Q M N @ U Y M? M V W Y B XG O@ W

;Q < G Q A B ^ < N] _ <]

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T

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Q G Q A U Y M? M V W Y M? M Q A < B XG O@ W 5

? ? ?

*/ !

#!

*)

# - Y

f(a) = g 1 (a, f

F(fils(a))

f F(s) = g ₂ (s, f(premier(s)), f F(reste(s)))

G Y M? M

G R J

Y M? M PQ A <

+

#

*)

$# Y

a = avide (fils(avide)

^F + !

"

).

s = nil (premier(nil)

reste(nil)

+ !

"

).

M M 7 = P 7 V > 7 QSR @ 9 :9 7 ?

@ 7 ] < ?@ 7

4 # % $E Y ! " # ) - * () * $*) $ $ %# Z)

# F) # %# () $* () [

! * $#

) D + Y

< Q Q YY M? M

→

^{@ < M}

< Q Q R Y J

Y M? M PQ A <

→

^{@ < M}

' () #! *) # Y

taille(a) = g ₁ (a, taille F(fils(a))

taille F(s) = g ₂ (s, taille(premier(s)), taille F(reste(s)))

M ^

A +

g 1 (a, z) = 1 + z g 2 (a, z 1 , z 2 ) = z 1 + z 2

* Y

taille(a) = 0

est arbre vide(a) taille(a) = 1 + taille F(fils(a))

taille

F(s) = 0

liste

vide(s) taille

F(s) = taille(premier(s)) + taille

F(reste(s))

6 > 7 QSR @ 9 :9 7 7 89 T: 6

;Q < M N < Q Q B ^ < N] _ <]

C

V W

DE J R T

Q P V Wb < Q Q RG J

Q P

@

< Q Q RG Q B ^ < N] Q _ <]

V W

D> U UM V W < Q Q > < Q Q RG M I I

Q < Q Q U Y M? M V W @ < B XG O@ W

Q < Q Q RG U Y M? M Q A < V W @ < B XG

O@

W M a P ? ? ?

4 # % $E Y ! " # ) - * () * $*) $ $

F % $ $) #

* ! ) , Z) F) # %# () $* () [

! * $#

) D + Y

@ A Q YY M? M

→

^{Y @ A}

@ A Q R Y J

Y M? M PQ A <

→

^{Y @ A}

' () #! *) # Y

ensval(a) = g ₁ (a, ensval F(fils(a))

ensval F(s) = g ₂ (s, ensval(premier(s)), ensval F(reste(s)))

M c

A + Y

g 1 (a, z) = {racine(a)} ∪ z g 2 (a, z 1 , z 2 ) = z 1 ∪ z 2

* Y

ensval(a) = ∅

est arbre vide(a)

ensval(a) = {racine(a)} ∪ ensval F(fils(a))

ensval

F(s) = ∅

liste

vide(s)

ensval F(s) = ensval(premier(s)) ∪ ensval F(reste(s))

, # * * Y + $

# $-

*

@ A Q

A [ ^

6 78 9 :9 7 8 : 9 7 8 Q 7 9 7 ] 79 ] 7 U T X

# %# % # # */ # */ ) # %# % # D + D

! () * D + @ < M [ ([

) 1 # %# B ) */ Z) @ * () #

- ! # ) # ) ! B ) , L @

) 1 # %# # Z) @ * () #

# * ) +! # ) # L@ ^ N ? P

b b b b b

E E E E ^ 5

7 ] 79 ] 7 QSR @ = 7 > Q 8 @ T

4 ) # # */ # */ # ) ! $

!

) .

Y

[

+ # # * $ # * H

0 [ .

*/

# */ #

$ " $ #

) B ) */ [

;Q < M N M N] M N] J

T P B ^ < N] _ <]

C

V W G Q A

D C M? J M T

G \ T

G P V W G B M <] @ < M O

Q A G X M <] @ M N] M N] J

T

G \ P

Q A M N] M N] J

T

G P I I

Q M N] M N] U Y H Y ? @ V W ? F FQ B XG

O@

W ^ 9 < = P 7 > Q 7 9 7 ] 79 ] 7 Q 8 @ T

! # ) . *

n

^{! $! [}

* + $, ! $ $$) # * *

O(1)

^[

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+ # *

O(n)

^[

^ ^

6 78 9 :9 7 8

) # %# 3 1 # ! () ! D + [ ([

[ */ () Z) ! () ! + # ) $

1 ≤ k ≤ 3

GF! $

!

[x 1 , . . . , x _k ]

^[ ([

x ₁ ≤ . . . ≤ x _k

0 [ ) Z) ! () ! + #

[x 1 , . . . , x k ]

^*

k + 1

^{" $}

f 1 , . . . , f _k+1

^[ (

) $ ! $!

f i

^{- ! # ) # ) !} B ) , L

x i

C + ) #

1 ≤ i ≤ k

J

) $ ! $!

f i

^{# * ) +! # ) # L}

x i−1

C + ) #

2 ≤ i ≤ k + 1

J

2[

) $

-

) $$ )

) ^ N@ 7 ?@ 7 8 7 7 =P 7 8

b I

b

D

I

b I

b I

b

D

b

b

b ^ 7

7 ] 79 ] 7 QSR @ = 7 > Q 8 @ 9 :9 7

4 ) # # */ # */ # ) ! $

! $

) # %# 0 [ 2[ 3

Y

[

+ # # * $ ! $! $ # * H

0 [ . */ # */ # $ % " $ $ $ +# E # * + # [

, # * * Y + $ # $- * M N] M N] A [ ^ 9 < = P 7 > Q 7 9 7 ] 79 ] 7 Q 8 @ 9 :9 7

! # ) # %# 0 [ 2[ 3 *

n

^{! $! / ) ) # ] J P[}

% # - ! # ) # Y $

F # %# 0 [ 2[ 3 * ) () )

! $! */ () )

H $#

2 ^{h ( a )} − 1 ≤ n

% # ) +

! # ) # Y $

F # %# 0 [ 2[ 3 * # ! $!

*/ () )

H $#

n ≤ 4 ^{h ( a )} − 1

2 ^{h ( a )} − 1 ≤ n ≤ 4 ^{h ( a )} − 1

2 ^{h ( a )} ≤ n + 1 ≤ 4 ^{h ( a )}

log ₄ (n + 1) ≤ h(a) ≤ log ₂ (n + 1)

^

* + $, ! $ $$) # * *

O(1)

^[

* + $, ! )

+

# *

O(log(n + 1))

^[

^ P P P

A ) $ $ ! "

) *

$

F

% $ GF # %#

A ) $ $ ! "

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#

GF) ) + $

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)

#

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*

K ) +# +#

! ! [

7 =P 7 V

4 # +#

! ! L ! # # Y

∀

^{Y ? @}

b, miroir(miroir(b)) = b

4 # ) Y 4 # ) * ) #

b

% Y

b = avide

^[

miroir(miroir(avide)) = miroir(avide) = avide

) * - Y

b = abr(m, a 1 , a 2 )

^[

miroir(miroir(b)) =

miroir(miroir(abr(m, a 1 , a ₂ ))) = miroir(abr(m, miroir(a 2 ), miroir(a 1 ))) = abr(m, miroir(miroir(a 1 ), miroir(miroir(a 2 )))) = h.r

abr(m, a 1 , a 2 ) =

b

5

P

4 # +# ! ! L ! # # Y

∀

^{C? @}

b, peigne(b) → prof(b) = nb

internes(b)

A > \@ $ +# +# ! ! Y

peigne(b) = peigne gauche(b) ∨ peigne droit(b)

peigne

gauche(avide) = true peigne

gauche(noeud(a 1 , a 2 )) = peigne

gauche(a 1 )∧

est

arbre

vide(a 2 ) peigne droit(. . .) = . . .

M 4 # ) Y 4 # ) * ) #

b

% Y

b = avide

^[

prof(avide) = 0 = nb internes(avide)

) * - Y

b = noeud(a 1 , a 2 )

^[

peigne(noeud(a 1 , a ₂ ))

+ $()

peigne gauche(noeud(a 1 , a ₂ ))∨peigne droit(noeud(a 1 , a ₂ ))

)

+ + + # B

! ! # $ !

peigne

gauche(noeud(a 1 , a 2 ))

^

F L # Y

peigne

gauche(a 1 ) ∧ a 2 = avide

$# Y

prof(noeud(a 1 , a 2 )) = 1 + prof(a 1 )

nb

internes(noeud(a 1 , a 2 )) = 1 + nb

internes(a 1 ) + 0

peigne gauche(a 1 )

^{H $#} + # / [ #[

prof(a 1 ) = prof(a 1 ) = nb internes(a 1 )

^*

prof(b) = 1 + prof(a 1 ) = 1 + nb internes(a 1 ) = nb internes(b)

^[

O P P U P P X

R

^{% $ ! %# % $ $ #} +# +

$$ [

> < F

% $

F prop

^{- # ) $ ) * $*) $}

+# + $H $ + $) +

% $ [ ([

R ⊆ F prop

A ∈ F prop

^{H $#}

¬A ∈ F prop

^[

A, B ∈ F _prop

H $#

∨(A, B)

H

∧(A, B)

H

→ (A, B) ∈ F _prop

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