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Mirkowska, ASD_08 Drzewa binarnych poszukiwaD O = ALGORYTMY I STRUKTURY DANYCH WYKAAD 08 Drzewa binarnych poszukiwaD Gra|yna Mirkowska PJWSTK, 2004&C  Plan wykBadu Drzewa BST Wstawianie Koszt utworzenia drzewa Usuwanie elementu z drzewa BST Sortowanie z BST Drzewo AVL Rotacje Koszty operacji b S  S  C& Sprawdzian Data 6 grudnia, sobota Godz. 10-12 sale aula + A1 + A2l E ( R/!Wstawianie elementu do drzewa BST "2PrzykBad 3Algorytm wstawiania 4Koszt utworzenia drzewa BST 8Koszt utworzenia c.d. )Operacja usuwania elementu 9Usuwanie - ilustracja 1 ( Usuwanie - ilustracja 2 :Usuwanie - ilustracja 3 ;(Zastosowanie: wyszukiwanie i sortowanie )*(Drzewo wywa|one AVL <6Obliczanie wag wierzchoBkw = Operacje na AVL +Rotacja w prawo @#Rotacja w lewo ?"Podwjna rotacja w prawo A$Podwjna rotacja w lewo B%6Ile rotacji trzeba wykona? D'PrzykBad ,Koszt operacji w AVL   ` @ ff3Ιd332z` @ ff3Ιd332z` 999MMM` fffPP3f>?" dd@ ?4Zd@ d " @ ` n?" dd@   @@``@n?" dd@  @@``PV    @ ` ` p>>  K0  D(  F   `  0PPf  c 6AminispirlB  <g   HD 1Ȝ? p`  TKliknij, aby edytowa styl tytuBu z Wzorca++  c $ p  Kliknij, aby edytowa style tekstu z Wzorca Drugi poziom Trzeci poziom Czwarty poziom Pity poziom,  c  c $ p   X* 2    c $T     Z*(2    c $H    Z*(2 XB   0D`Z  BsZ޽h))?? @ ff3Ιd332z Notatnik&  K0 @ <(   FF       XA StationeryPP`   S 0AminispirH  < ?^~  ALGORYTMY I STRUKTURY DANYCHKliknij, aby edytowa styl tytuBu z WzorcaGG  c $ 3    ZKliknij, aby edytowa styl podtytuBu z Wzorca..  c $H ^#  X* 2f   c $4 ^~#   Z*(2f   c $ ~#  Z*(2f Z  BsZ޽h))?? @ ff3Ιd332z| 0 ( F0E,   0 P    T*   0     V* d  c $ ?    04  @  Kliknij, aby edytowa style wzorca tekstu Drugi poziom Trzeci poziom Czwarty poziom Pity poziom*  a  6 `P   T*   6 `   V* H  0޽h ? ̙3380___PPT10.G@My K0 0(  l  C  ^~  l  C к 3    H  0޽h ? @ ff3Ιd332zy___PPT10Y+D=' = @B +  K0 P@4(  @l @ C x' p`    @ S    " PpH @ 0޽h ? @ ff3Ιd332zy___PPT10Y+D=' = @B +<  K0 $(  r  S  p`   r  S p  H  0޽h ? @ ff3Ιd332z80___PPT10.GM_!#  K0   `l (  l  C <% p`   B  3 ZwG UNd)?MetodaArial Black$ k !2P  ,$D  0V  6'0P w,$   0 , insert : BST ET BSTF 2 &  <(3J,$  0 |6Rozpoczynajc od korzenia drzewa D przegldamy wierzchoBki tak, jak w operacji wyszukiwania: Je[li znajdziemy wierzchoBek z etykiet e, to wynikiem operacji jest dane drzewo D. Je[li e nie jest etykiet drzewa D, to tworzymy nowy wierzchoBek z etykiet e i dowizujemy go 1. jako lewego syna wierzchoBka v takiego, |e e< et(v) i LP jest puste lub 2. jako prawego syna v, gdy et(v)< e, oraz PD jest puste.  2   <)p,$  0 TZadanie Do zbioru reprezentowanego przez drzewo D doBczy element e, o ile nie nale|y on jeszcze do etykiet drzewa D.4yq yH  0޽h ? @ ff3Ιd332z___PPT10+ *[D' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*y%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*yD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*yD' =%(D+' =%(D' =A@BBBB0B@B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' K=+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' K=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ +C  K0 4,p%%( ? l  C |C p`   2  BE@,$D  0 O6 z p@   P`0 P,$D  02  BJP@ O6 2  BhOp `  O5 lB B <D0   ZTI#vd @Pp,$D  0 E5     `XHjJIvd @ p,$D  0 E9    f\HLJI vd @@,$D  0 F12 z  0`     ,$D  02   B4a0 O6 2   Be p 0  O5 lB B <D p2  B|i `  O9 lB  <Dz 0   ` 0 ,$D  0N  0`    2  Bm0 O6 2  Bq p 0  O5 lB B <D p2  B0v `  O9 lB  <D2  B z@` 0  P12 lB  <D p    f~G4HLJIvd @ @ ,$D   0 E8 z  @   pp,$D   0T  0`  #  @p 2  B 0 O6 2  B4 p 0  O5 lB B <D p2   B `  O9 lB ! <D2 " B   P12 lB # <Dp  2 $ B̖   E8 lB %B <Dp p H  0޽h ? @ ff3Ιd332z"**___PPT10)+PwD(' = @B D(' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(#/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+  ++0+  ++0+ +)  K0    ( s l  C  p`     <@0,$  0 \{ bool := false; x:= root ; while not bool { if x.et= e then bool := true else if (e < x.et) then if ( x.lewy <>null) then x := x.lewy else y := New node(e); x.lewy := y; bool := true fi else {//analogicznie dla prawego //poddrzewa} fi fi }} 2b      ! E<   S W3yd?LematImpact@,$D  0N  S ZwG UNd)?insertArial Black$ k !2p ,$D  0  < ,$  0 `Algorytm insert zatrzymuje si dla wszystkich danych pocztkowych. Otrzymane w wyniku drzewo ma w zbiorze swoich etykiet e.} 2|& n  S W3yd?Koszt :ImpactP p X,$D  0   < 0,$  0 mA(n) = O(lg n) 2&H  0޽h ? @ ff3Ιd332z___PPT10+zYD' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*}%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*}D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*}Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ +])  K0   8 (  >\@ l  C  p`     <,$  0 UWAGA Koszt utworzenia i struktura drzewa zale| od kolejno[ci wkBadanych elementw.&U 2P U  <` ,$  0 b W(n) = O(n2)& 2     tAA)Papeteria  ,$D  0 "Zredni koszt utworzenia drzewa BST o n wierzchoBkach wynosi O(n lg n), por. uzasadnienie.&Z 2G&@  TGHv/ @ ,$D  0 Najgorszy przypadek = wkBadane elementy tworz cig uporzdkowanyDD DL  S Z wG UNd)?LematArial Black$ k !2 hr ,$D  0z PP   PP ,$D  02   B E9 lB  B <DP` 2   B`   E8 2   B `  E7 lB  B <D0 @ 2  BP P  E6 lB B <D  H  0޽h ? @ ff3Ιd332z-%___PPT10+YD ' = @B D' = @BA?%,( < +O%,( < +D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*U%(D' =-o6Bbox(out)*<3<*UD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ ++0+ +x5  K0 e ]  (  E\p l  C   p`   >  < j,$  0 PNiech wkBadane do drzewa elementy bd permutacj liczb 1...n i niech prawdopodobieDstwo tego, |e i-tym elementem jest k bdzie takie samo dla wszystkich k=1,2,...n. 2     # @ ,$D  02  B Ei lB B <Dp lB  <D  B   PLD R   B@  PPD    c $A ??008 $D  0   c $A ?? 0P 8 $D  0B  BD  ,$D  0B  BD ` ,$D  0z  b $APapeteria 0 ,$D  0 fKa|dy wkBadany element jest porwnywany z korzeniem4 23 4  c $A ?? ` 8 $D  0  b)APapeteria ,$   0 <Hipoteza : A(i) k * i *lg i J 2&H  0޽h ? @ ff3Ιd332z''___PPT10{'+YٲD&' = @B Dr&' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(!/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(&/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ +  K0 , $ ( q@G@ l  C C p`     t WԔ?delete(D, e)TECHMathd  s *EP ,$   0 ,delete : BST Et BSTl 2 B  3 ZwG UNd)?MetodaArial Black$ k !2`0 ,$D  0  0TNp0 ,$D  0 "(1) Znajdujemy wierzchoBek x o etykiecie e stosujc algorytm member i zapamitujemy jego ojca y. (2) Dalsze postpowanie zale|y od liczby nastpnikw x: - Usuwamy wierzchoBek x, je[li jest on li[ciem. - Zastpujemy wierzchoBek x jego nastpnikiem, je[li x ma tylko jednego syna. - Zastpujemy etykiet wierzchoBka x, najmniejsz etykiet w jego prawym poddrzewie (lub najwiksz w jego lewym poddrzewie), a wierzchoBek o tej etykiecie usuwamy z drzewa, stosujc zasad (1) lub(2).  2&=H  0޽h ? @ ff3Ιd332z ___PPT10 +A<DP ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BBBB0B@B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' K=-o6Bwipe(up)*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D|' =%(D$' =A@BB BB0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+p+0+ ++0+ +x@  K0 4,!0@( x l  C _ p`   L  0la`,$  0 r1 Przypadek : x nie ma synw, tzn. jest li[ciem (rz(x)=0): 2: :   s *lf @`,$D  0 : Usuwamy wierzchoBek x.( 2 ol p @  0p @ ,$D  02  <kh  Oy R  <p@ p @  FPD fB   6Dh bX   s *,$D  0z @    0p ,$D  02  6t` Oy 2  6y@ Ox R  6~@  FPD `B B 0D 0`B  0D0XB  0D0 0 0 l ` '  "` ' ,$D  02  <8 Oy 2  <`8  Ox R  <` '  FLD fB B 6D b8fB   6Db ! s * ,$D  0ol p'  )p' ,$D   02 $ <АH Oy R & <p'  FLD fB 'B 6DbH^z  @ , O(P ,$D  0fB * 6DԔ  fB +B 6DԔ @d  @ -# O(P ,$D  0fB . 6DԔ  fB /B 6DԔ @H  0޽h ? @ ff3Ιd332z++___PPT10++<\D +' = @B D*' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*:D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*:D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-o6Bbox(out)*<3<*,D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*"D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*"D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(-/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-o6Bbox(out)*<3<*-D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(2/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-o6Bbox(out)*<3<*!Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*)D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*)+p+0+ ++0+ +?4  K0 e]( (  l  C   p`   '  0`,$  0 32. Przypadek : x ma jednego syna, tzn. rz(x) = 1.4 24&)   s *0` 0 ,$D  0 : Usuwamy wierzchoBek x.( 2   s * P,$D  0l   #PP ,$D  02  <t88 Oy 2  <P   Ox R  < i  FPD lB B <DԔH lB  <DԔWfB B 6DԔ0 R   6  SLD(x) l 0  " ,$D  02  <@h Oy R  <|  KLD(x) lB B <Do `@lB  <Do0 R ! 6$0  FPD d  @ $# O(`,$D  0fB % 6DԔ  fB &B 6DԔ @P ' 0x) p,$  0 vPostpowanie jest analogiczne, gdy x ma tylko prawego syna.< 2< < ( 04. P  ,$D 0 Wy.lewy := x.lewy; 2 H  0޽h ? @ ff3Ιd332zj#b#___PPT10B#+R5DF"' = @B D"' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*#D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*#D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bbox(out)*<3<*$D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*"D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*"D%' =%(D' =%(Du' =A@BBBB0B%(E' =4 B`BPB`B?<*%('/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*(%(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'<%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*'<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*'<++0+ ++0+ ++0+' ++0+( +8  K0 xp) ( ^: l  C B p`   ~  s *0D @,$D  0  Zastpujemy wierzchoBek x jego bezpo[rednim nastpnikiem w drzewie .(M 2L M  s * P,$D  0&  0I`,$  0 23. Przypadek : x ma dwch synw, tzn. rz(x) = 2.3 23&(,l @@ P  p ,$D  02  <xO@ Oy 2  <T( Ox R  <X `  FPD lB  B <DԔplB   <DԔ lB  B <DԔR   <d]@  SLD(x) R  <`@pP  KPD(x) fB  6DԔ@2 % 0e@.  ,$D  0 Ez l  pp  'p p ,$D  0N @@ P    pp 2  6i@ Oy 2  6xn( Ox R  6r `  FPD fB B 6DԔpfB   6DԔ fB !B 6DԔR " 6v@  SLD(x) R # 6l{@pP  R PD (x) fB $ 6DԔ@Z2 & s *    ( 0 ,$D  0 Q Et(x)=Et(z) 2  = ) 0z 0 ,$D  0 A z := min(PD(x)); Et(x) := et(z); x.prawy:= delete(PD(x), et(z));B 2A&,  H  0޽h ? @ ff3Ιd332z##___PPT10#+F?HD"' = @B DB"' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*3D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*3D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-o6Bbox(out)*<3<*%D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*'D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*'D' =%(D' =%(D:' =A@BBB*B0B%(E' =4 B`BPB`B?<* %('/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-6B)barn(outHorizontal)*<3<*(D' =%(D' =%(D:' =A@BBB*B0B%(E' =4 B`BPB`B?<* %(,/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-6B)barn(outHorizontal)*<3<*)++0+ ++0+ ++0+% ++0+( ++0+) +<,  K0     (  l  C | p`     0p`,$D   0 |(Zadanie A Dany jest zbir n elementw nale|cych do pewnego uporzdkowanego zbioru. Zbada, czy dany element nale|y, czy nie nale|y do tego zbioru.$ 2    0Ȣ 0 ,$D  0 rZadanie B Dany jest zbir n elementw nale|cych do pewnego uporzdkowanego zbioru. Uporzdkowa elementy tego zbioru w porzdku niemalejcym.$ 2 f l  0Ȩ 0P ,$  0  w tablicy z u|yciem listy dynamicznej z u|yciem drzewa BST"D 2A DL  3 ZwG UNd)?RozwizanieArial Black$ k !2 `,$D 0l @ `   @`,$D  0   V`APapeteria@ @ 21.Zbudowa drzewo BST, 2.Odczyta jego wierzchoBki w porzdku inorder (infixowym)R 2R@>  @`   |Zw?sortowanieArial Black@ `|H  0޽h ? @ ff3Ιd332z  ___PPT10 +'XD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D6' =A@BBB%B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%barn(outVertical)*<3<*D' =%(D' =%(D4' =A@BBBB0B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B#blinds(vertical)*<3<*DR' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<* ,%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* ,D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* ,D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*,D%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*,DD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*,DD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %('/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* ++0+ ++0+ ++0+ +8  K0 !# d( ,mz3 l  C  p`     3 ZwG UNd)?DefinicjaArial Black$ k !20   0j,$  0 n(Powiemy, |e drzewo binarne jest wywa|one, je|eli dla wszystkich jego wierzchoBkw, wysoko[ci lewego i prawego poddrzewa r|ni si co najwy|ej o 1.  2 x  0p@z,$  0 Wywa|one drzewo BST nazywamy drzewem AVL (Adelson-Velskii-Landis)C 2C&+F  3 ZwG UNd)?PrzykBadArial Black$ k !2 ,$D  0z  @    @  ,$D  0T  0`  #  @p 2   Bp0 O6 2   B p 0  O5 lB  B <D p2   B `  O9 lB   <D2  B   P12 lB  <Dp  2  Bt   E8 lB B <Dp p  l `  ! p J,$D  02  H ` p  O6 2  Hp-   O5 rB B BD - 2  HLS  O9 rB  BDF @ S 2  HD  P12 rB  BDj : 2  HF   E8 rB B BDp  F lB B <D  2  B,  PZ  O2 2  B O3 lB   <D@ :@ " 0  ,$  0 R Drzewo AVL 2   # 0@ :,$  0 `A to nie jest drzewo AVL 2 H  0޽h ? @ ff3Ιd332z___PPT10{+|U=D' = @B D:' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*CD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*CD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*" %(D' =-o6Bbox(out)*<3<*" Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*!D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*!D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %('/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-o6Bbox(out)*<3<*#++0+ ++0+ ++0+" ++0+# ++  K0  &( x l  C ' p`   0  0)p:,$  0 VNiech w bdzie funkcj okre[lon na wierzchoBkach drzewa BST tak, |e w( x) = h(LD) - h(PD), gdzie LD i PD s odpowiednio lewym i prawym poddrzewem drzewa o korzeniu w x. 2 z `   @ : ,$D  02  B/` p  O6 2  B4-   O5 lB B <D - 2  B8S  O9 lB  <DF @ S 2  B=  P12 lB  <Dj : 2  BAF   E8 lB B <Dp  F lB B <D  2  B? PZ  O2 2  B J O3 lB  <D@ :@l   P  &  P ,$D   0  0P+0 P  S0" 2   0R@p 0  S0" 2   0V  0  S0" 2    0ZP @  S0" 2  ! 0_0  S2" 2  " 0d P p  T-1" 2  # 0h P  T+1" 2  $ 0lPP  ,$  0 ZUwaga Drzewo binarnych poszukiwaD D jest drzewem AVL wttw dla ka|dego xD.V , w(x) {-1, 0, +1}.Pa 2fA &5(H  0޽h ? @ ff3Ιd332z___PPT10+D$Dj' = @B D%' = @BA?%,( < +O%,( < +D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DR' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*&D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*&D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bbox(out)*<3<*$D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*$a%(D' =-o6Bbox(out)*<3<*$a+p+0+ ++0+$ +2g  K0 x,p, 8> +(  r  S T p`   B  0 :,$  0 member : AVL Et Bo insert : AVL Et AVL delete : AVL Et AVLG 2   N  B8c ,$D  0 < H  0< @,$  0 XWykonuje si tak jak na drzewach BST, ale...,- 2%f -  0l`  ,$D 0 DoBczenie nowego elementu do drzewa BST mo|e zwikszy wysoko[ jakiego[ poddrzewa!U 2U U  0`0 ,$D  0 Usunicie jakiego[ elementu z drzewa BST mo|e zmniejszy wysoko[ jakiego[ poddrzewa!X 2X X' l P @  %P @ ,$D  02   H$Z,$D   0 S6" 2   HPp@' ,$D   0 S5" B  B BDZ:,$D   02   HD3 M ,$D    0 S9" B   BD3,$D    02  Hd Z P ,$D    0 T12" B  BDM ,$D    02  H: @ ,$D    0 E8 B B BDM  ,$D   0B B BDP0 tc ,$D   02  Hк  ,$D   0 S2" l    $ : ,$D  02  Bп@p   S3" rB  BD  z ` `p  & ` p ,$D   0 ' 0T` I0 2  ( 0 0  J+1 2  ) 0,`   I0 2  * 000  I0 2  + 0P p  I0 2  , 0 P@  I0 2 z ` `p  - ` p ,$D  0 . 0` J+1 2  / 0P 0  J+2 2  0 0`   J-1 2  1 000  I0 2  2 0<P p  I0 2  3 0 P@  I0 2 vl   7 ,$D   0t 0   40  ,$D    02  HX0 p  O6 2  H4   O5 rB B BD  2  H#   O9 rB  BDF @# 2  HD c  P12 rB  BDj : 2  H   E8 rB B BDp   5 0@  F-1 2  6 0`   F 0 2 Gz @*  8   ,$D    02 9 Ho@],$D   0 [5* B :B BDo* ],$D   0 ; s *4p ,$D  0 F-2 2 Az @*  <   ,$D    0 2 = B;@],$D   0 [5* B >B BD;* ],$D   0H  0޽h ? @ ff3Ιd332zJ:B:___PPT10":+1(D8' = @B Dq8' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*GD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*GD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*-D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*-D' =%(D' =%(D+' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-6B'blinds(horizontal)*<3<*%D' =%(D' =%(D;' =4@BBB B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*&E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-o6Bbox(out)*<3<*&D' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(&/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bbox(out)*<3<*$D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(+/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-o6Bbox(out)*<3<*-D' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*D' =%(D' =%(D+' =4@BBB B%(E' =4 B`BPB`B?<* %(5/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-6B'blinds(horizontal)*<3<*7D' =%(D' =%(D;' =4@BBB B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*8E' =4 B`BPB`B?<* %(:/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-o6Bbox(out)*<3<*8DR' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<* %(?/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<D_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(E/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*;D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*;+P+0+ ++0+ ++0+ ++0+ ++0+ ++0+; +/  K0 0"q(  l  C <7 p`   dB  <D @` dB  <Dp p l  ,$D 02  69P`@P EA 2  6$>p EB fB B 6DԔ@`fB  6DԔ@  6Ap  IZ     6dE@ _X *(    6 Gp@  IY  fB  B 6DԔ` P@fB   6DԔ@`@  <O1 R+2    <S0q  R+1  l p  "p ,$D  02  <0X EB 2  <U@0p EA lB B <DԔ0@lB  <DԔ  <`  IZ    <]p  _X *(   <$h@@`  IY  lB  <DԔ0   BDl`4 Q0    Bp Q0  fB B 6DԔ   HG:fvd @ p,$D  0  6u@P P 0 ,$D 0 T Po rotacji    ! 6hzf ,$D   0 E* H  0޽h ? @ ff3Ιd332zYQ___PPT101+)D' = @B D`' = @BA?%,( < +O%,( < +DR' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(DH' =A@BBB B0B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*!E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-o6Bbox(out)*<3<*!DL' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(D-' =4@BBB*B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-6B)barn(outHorizontal)*<3<*"+p+0+  ++0+! +E.  K0 @ "(  r  S  p`   dB  <Dp p   BЊ @T@ R-2  z p  A p ,$D  02  6P EB 2  6p@0p EA fB B 6DԔ0@fB  6DԔ  6<  IZ    6p  _X *(   68@@`  IY  fB  6DԔ0   <L`4 Q0    < Q0  fB B 6DԔ  @ HG:fvd @`@,$D  0  BLG8H@P ` ,$D 0 T Po rotacji   l  A,$D  02  <HP`@P EA 2  <lp EB lB B <DԔ@`lB  <DԔ@   <p  IZ     <l@ _X *(    <p@  IY  lB  B <DԔ` P@lB   <DԔ@`@  Bh  R-1    6Df ,$D   0 E* dB ! <D @`  " 0lpPj,$  0 h"Pojedyncza rotacja w lewo wzgl. B# 2# #H  0޽h ? @ ff3Ιd332z___PPT10+3D ' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D+' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DL' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(D-' =4@BBB*B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*"#%(D' =-o6Bbox(out)*<3<*"#+p+0+ ++0+" + ?  K0 P+/ (  r  S D p`   dB  <D @` dB  <Dp p   TG7HJI4fvd @ @,$D  0  6P P 0 ,$D 0 U Po rotacji    l `  %p0 ,$D  02  <P`@P EA 2  <p EC lB B <DԔ@`lB   <DԔ@   <@` p  IU     <,@  IX     <P p JY * lB  B <DԔ` P@lB  <DԔ@  BD R+2    B\0D  R-1  2   6< p` EB  ! 6 `   GZ  fB "B 6DԔ fB # 6DԔ0   $ 0`  L+1 2  & 6\ @,$D   0 E*  ' 08` FW lewo wzgldem A i w prawo wzgl. C$ 2$ $ (@ TG7HJI4fvd @0,$D  0 l  $  / $ ,$D  02  < EC 2  <@0p EB lB B <DԔ@lB  <DԔ  < $  IU    <' @  IX    <+@p  GZ  lB  <DԔ0P  B/ Q0    B4$ R-1  lB B <DԔ0 @2 ) 690  EA  * 6=@  IY  fB ,B 6DԔp`@fB - 6DԔ@ . <A  Q0  H  0޽h ? @ ff3Ιd332z""___PPT10s"+D!' = @B D!' = @BA?%,( < +O%,( < +D' =%(D' =%(D+' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-6B'blinds(horizontal)*<3<*%D_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*&D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*&DL' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*(D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*(D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*(DL' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(D+' =4@BBB B%(E' =4 B`BPB`B?<* %(%/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-6B'blinds(horizontal)*<3<*/+p+0+ ++0+& +:  K0 `*1 b(  dB  <D @` dB  <Dp p   TG7HJI4fvd @ P,$D  0  6HUP P 0 ,$D 0 U Po rotacji     0Y` FW prawo wzgldem A i w lewo wzgl. C$ 2$ $ @ TG7HJI4fvd @`@,$D  0( l  0  1 0 ,$D   02 B <^ EC 2 B <Wp EB lB  <DԔlB  B <DԔ@0 !B < g   IU   "B <d@0  IX   #B <hn@p  GZ  lB $B <DԔ %B BDr  Q0  lB ' <DԔ@2 (B <v EA  )B <lt0@P  JY * lB * <DԔp@lB +B <DԔ` @ ,B B~ Q0   .  |0e0e 1Ȝ? p`    l p 0  0 p0 ,$D  02  B <`P EA 2  B <p EC lB   <DԔ@ `lB  B <DԔ`@  B <Ppp  IU   B < @0  IX   B <h`  JY * lB  <DԔ p@lB B <DԔ  B B @ R-2  2 B <p` EB  B <О  p  GZ  lB  <DԔ0 lB B <DԔ  B 6 L-1 2   6` ,$D   0 E*  /B <`P  R+1  H  0޽h ? @ ff3Ιd332zsk___PPT10K++B#style.visibility<*0%(D' =-6B'blinds(horizontal)*<3<*0DL' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*DL' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DR' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*1D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*1+8+0+ +:  K0 p(  r  S L p`   v  00 ,$  0 Je[li wkBadamy element do drzewa AVL, to musimy wykona co najwy|ej 1 rotacj.O 2O O  0ܻ@ ," ,$  0 4Je[li usuwamy element z AVL, to mo|e si zdarzy, |e bdziemy musieli wykona tyle rotacji ile jest poziomw w drzewie.x 2x x  BDG @ U 0U 0 T pokaz   0,$ 0 6Koszt 1 rotacji jest staBy! 2f   0 J=,$  0 Przy usuwaniu jednego elementu z AVL o n wierzchoBkach, liczba rotacji wynosi co najwy|ej lg n.a 2a&[  0  V @,U 0U 0H  0޽h ? @ ff3Ιd332z___PPT10+D' = @B Dc' = @BA?%,( < +O%,( < +D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*O%(D' =-o6Bbox(out)*<3<*OD_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*x%(D' =-o6Bbox(out)*<3<*xD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ ++0+ +l  K0 FFyPF(  r  S  p`    l - S  2- S ,$D  02   < P  H11 2   <M @  G9 2   <P G7 2   <- G6 TB  c $DZM m2  <lM  G8 2  <t]   H10 2  <#   H13 2  <Pm  S  H12 2  <43  S  H14 TB B c $DjM }TB  c $D  PTB B c $D=TB  c $DjTB  B c $D @ TB ! c $Dm P TB "B c $D P TB # c $D~ `  0 0   H-1 2  1 0D   H-1 2 S8 M  6M @ MZ MZ2  < W G3 2  < G" G2 2  <"g G4 2  <` G1 2  <Z/ G5 TB  c $DTB B c $D}t`TB  c $D "TB B c $DmW"  08t H+1 2   0lMZ! 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