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TjJ?"PFPZ  @  f  8c?"p  ,$D 0 A X(M(S()))  H  0޽h ?  $129:;:<=>? 13  ___PPT10 +˝D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-o6Bdissolve*<3<*"D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-o6Bdissolve*<3<*@D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*++0+ ++0+" ++0+@ +     -D@ (  Dr D S 0 `    D # l3 8c?"M- 9t =  l @   D@  ,$D 0 D  fD+ 8c?" 0 = root of t  B DB  `D8c?"@   D # l(< 8c?" L 0 <subtrees   F ` @  !D ` R "D T 8c?"  R #D T 8c?" U  $DB T8c?"  %D T8c?"} R &D N 8c?"` @ R 'D T 8c?"   (DB T8c?"( 2 )D Z 8c?" *D  f@F 8c?"@ ]  ;. . . B +D  fD8c?"p P B ,D  fD8c?"p @ B -D  fD8c?"p ` H D 0޽h ??` )D"D$D )D#D%D )D'D(D 3333    PP(  Pr P S J `    P S ,K  "<$ 0  H P 0޽h ? 3333~v___PPT10V.+NwnD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Pz%(D' =-o6Bdissolve*<3<*PzD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Pz%(D' =-o6Bdissolve*<3<*Pz+8+0+P +7    0,iH@(  Hx H c $_ `  _ f8  9H   H  f0[ 8c?" a-Processing non-smallest incoming values of t:..   H  f_ 8c?"  ])Processing smallest incoming values of t:**R H Z 8c?"pp@ 2 H  f 8c?"`PR H Z 8c?"@>D R H Z 8c?"@0oD  H@  `jJ?"58  H  `jJ?"58 +H # ld 8c?" 8#t =R @H T 8c?"@D  AH@ TjJ?"X!X8 BH  fi 8c?"96Y  9. . .l pb bHpb,$D 0B EH  fD8c?"p0 FH  fXm 8c?"^, d.1: display root and  ( 2 GH  ` 8c?"`ZPJ JH  fq 8c?" Bb 5(Pl   cH  ,$D 0B NH  `D8c?" p B PH  `D8c?"p p R QH T 8c?"  RH Zv 8c?"p j  N2: remove subtree from t\l  *  dH * ,$D 0B SH ZD8c?" p @p B TH  `D8c?"@@p R UH T 8c?" VH Z{ 8c?" p *  `*3: display subtree by calling Display_Tree++dl  0  eH 0 ,$D 0B WH  `D8c?"0 0 B XHB  `D8c?" 0 B YH  fD8c?"   ZH Z 8c?"  P4: place subtree back in tz @BF  iH B@F ,$D 0 [H # l 8c?" F  9. . . ]H # lԃ 8c?"BFb 9. . ., ^H  `̑ 8c?"@@P  .repeat steps 2-4 for each of the subtrees of t//!z 0B J  hH B0 J ,$D 0 \H  f 8c?"Bb 5) _H  ` 8c?"0P J  R5: display  ) B `H # lD8c?" ; aH  f 8c?"0;,$D 0 c/No special handling of smallest incoming values00H H 0޽h ??`HHHHH HH@HAH 3333___PPT10+yD' = @B D;' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*bH%(D' =-o6Bdissolve*<3<*bHD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*cH%(D' =-o6Bdissolve*<3<*cHD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*dH%(D' =-o6Bdissolve*<3<*dHD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*eH%(D' =-o6Bdissolve*<3<*eHD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*iH%(D' =-o6Bdissolve*<3<*iHD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*hH%(D' =-o6Bdissolve*<3<*hHD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*aH%(D' =-o6Bdissolve*<3<*aH+8+0+aH +*  @<&(  <r < S h `   r < S < p   <  f 8c?"]0,$D  0 " display root and  ( for each of the subtrees of t { remove subtree from t display subtree by calling Display_Tree place subtree back in t } display  ) &0n<ZP-'!H < 0޽h ? 3333%%___PPT10r%+n*D%' = @B D$' = @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<*<;D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<;A%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<;AD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<;AD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<A_%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<A_D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<A_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<*<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<*<+8+0+< +"4  *"@T(  Tx T c $8% `   x T c $d' p   T  fh) 8c?"d,$D  0 ( object Integer pos; outs << t[current] <<  ( ; while (pos < t.Number_Of_Children ()) { object Tree_Of_Integer subtree; t.Remove (pos, subtree); Display_Tree (subtree, outs); t.Add (pos, subtree); pos++; } outs <<  ) ;f2)H T 0޽h ? 3333..___PPT10x.+n*D$.' = @B D-' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*T8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*T8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T8b%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*T8bD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*T8bD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Tb%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TbD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TbD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*T+8+0+T +2*  ))`8Cp$ ~((  px p c $I `     p # lJ 8c? " a-Processing non-smallest incoming values of t:.. p # lO 8c? " ])Processing smallest incoming values of t:**R p Z 8c?"pV 2 p  f 8c?"`PR p Z 8c?"@TD R  p Z 8c?"@FD   p@  `jJ?"58  p  `jJ?"58  p # lU 8c?"0; 8#t =R  p Z 8c?"@D  p@ ZjJ?"X7n8 p # lY 8c?"9LY  9. . .B p  fD8c?"`  p  f] 8c?"~  d.1: display root and  ( 2 p  ` 8c?"Z J p  fPb 8c?"Bb 5(B p  fD8c?" p B p  fD8c?"p Tp R p Z 8c?" @  p  `i 8c?"p rj  N2: remove subtree from tB p  `D8c?"p ` p B p  fD8c?"p R p Z 8c?"^ p  `n 8c?" 0 J  `*3: display subtree by calling Display_Tree++B  p  fD8c?"  B !p@  `D8c?"  B "p  fD8c?"`   #p  ` t 8c?" B}  K4: place subtree in tmp &p # ly 8c?"B^b 9. . .$ 'p  `} 8c?":  .repeat steps 2-4 for each of the subtrees of t//! )p  f  8c?"Bb 5) *p  `  8c?"UO R5: display  ) B +p  fD8c?"00`; ,p  fd 8c?"0;,$D 0 c/No special handling of smallest incoming values00R /p Z 8c?"  2 0p  f 8c?"  R 1p Z 8c?" 0 R 2p Z 8c?"   3p@  `jJ?"   4p  `jJ?" ]  5p # l 8c?"  9= tmpR 6p Z 8c?" 0r  7p@ ZjJ?"   8p # l 8c?" \  9. . .B 9p  `D8c?"0pB :p  `D8c?"pp`B p  fD8c?"`   ?p Z 8c?"   A 6: swap rootsB @p  `D8c?" @B Ap ZD8c?"@@B Bp  fD8c?"P @ Cp Z` 8c?"@: E7: swap t and tmpH p 0޽h ?opp pp p pp pp0p1p3p0p2p4p0p6p7p 3333(4  0(pt(  tx t c $ `   x t c $t p   t  f 8c?"H,$D  0  display root and  ( for each of the subtrees of t { remove subtree from t display subtree by calling Display_Tree place subtree in tmp } display  ) swap roots of t and tmp // MUST DO THIS!! swap t and tmp&0n<Zb-'6&H t 0޽h ? 3333..___PPT10x.+FD$.' = @B D-' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*tD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*tD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t;%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*t;D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*t;D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t;A%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*t;AD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*t;AD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*tA_%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*tA_D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*tA_D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t_%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*t_D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*t_D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*tD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*tD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*tD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*tD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*tD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*tD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*tD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*tD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*tD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*t+8+0+t +=9  xR(  xx x c $p `   x x c $| p   x  f 8c?"d,$D  0 B object Tree_Of_Integer tmp; outs << t[current] <<  ( ; while (t.Number_Of_Children () > 0) { object Tree_Of_Integer subtree; t.Remove (0, subtree); Display_Tree (subtree, outs); tmp.Add (tmp.Number_Of_Children (), subtree); } outs <<  ) ; t[current] &= tmp[current]; // MUST DO THIS!! t &= tmp;fa9'H x 0޽h ? 3333#33___PPT102+FD2' = @B Db2' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*x D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*x D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x ?%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*x ?D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*x ?D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x?g%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*x?gD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*x?gD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*xg%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*xgD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*xgD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*xD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*xD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*xD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*xD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*x D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*x D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*x D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*x D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x!%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*x!D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*x!D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x!S%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*x!SD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*x!SD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*xSa%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*xSaD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*xSa+8+0+x + 0 zrP  (   X  C    r  S  @   H  0޽h ? ̙33 0 |F(  |X | C     | S i @   H4These are math operations that are used in the specsH | 0޽h ? ̙33 0 zr (  X  C    r  S t @   H  0޽h ? ̙33 0 O(  X  C      S (R @   Q=This is another example of visualizing recursive process. 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We can go over it if there is time, or simply skip it and the following slides.H  0޽h ? ̙33rPC%3e= R_g ,k l~h]$C*cE>aDmX'.,m$L%( H/ 0|Oh+'0U hp  $ 0 <HPBinary_Tree ComponentP Paolo Bucci8C:\WINDOWS\Application Data\Microsoft\Templates\321.pot Paolo Bucci26lMicrosoft PowerPointn D@*40b@s@x;gGSg  )'    """)))UUUMMMBBB999|PP3f333f3333f3ffffff3f̙3ff333f333333333f33333333f33f3ff3f3f3f3333f33̙33333f333333f3333f3ffffff3f33ff3f3f3f3fff3ffffffffff3ffff̙fff3fffff3fff333f3f3ff3ff33f̙̙3̙ff̙̙̙3f̙3f333f3333f3ffffff3f̙3f3f3f333f3333f3ffffff3f̙3f3ffffffffff!___www64'A x(xKʦ """)))UUUMMMBBB999|PP3f3333f333ff3fffff3f3f̙f3333f3333333333f3333333f3f33ff3f3f3f3333f3333333f3̙33333f333ff3ffffff3f33f3ff3f3f3ffff3fffffffff3fffffff3f̙ffff3ff333f3ff33fff33f3ff̙3f3f3333f333ff3fffff̙̙3̙f̙̙̙3f̙3f3f3333f333ff3fffff3f3f̙3ffffffffff!___www!+$+!B$%$!+$+!+$B$+!%$B$+!+$B!+!!$B!+!!$B!H!!!B!B!!!B!HB!!!!B!B!!!!!!!!!H!B!B!H!HB!HB!H!BH!!!!!B!!H!!!B!!!B!!!!H!H!!!B!!!!B!!B!H!B!!!B!!B!B!B!B!B!B!9!B9!B!!B!B!B'N!!B!B!B!B!B!B!BHBB!B!B!B!9!B!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!H!!!!!!!!!!!!!B!H!BH!B!H!!B!H!B!H!B!H!B!!B!H!B!!!H!!B'N!H!H!B!H!B!H!B!!!!!'B!!'H!!B!!!!H!!!!!!!!!!!B!!!!B!!'!!!B!9!!!B!B!B!B!B!!B!B!H!BHB'B!BBHB!B!!B!B!B!B!!'B!B!BB!B!B!!!!!!!94!!!!!!!!!!!'H'H!!!!4!!!!!!!!!!!!!!BH!B!H!B!H!BHB!HHBH!!9!!N!H!!B!!B!H!B!H!!B!!!B!B!B!!!B!B!!!!B!!!!B!!'B!!B4!!B!!!BB!B!!B!B!BB!B!B!BB!B!B!N!B'BB!BHB!B!!!9!9!!!!!!!H!!!!!!4!!!!H!!!!!!!!!!!!!H!B!!B!N!B!N!B!HHH!H!H!H!BH!B!H!B!H!HH!B!H!B!!B!H!B!!!!H!B!4!'!!B!!!!9!B!!!!!!!B!!4!!!!B!H'!!H!B'!4!!B!!!!!!!B!!!!B!B!B9!B!B!B!B!BB!B!B!B!B!B!B!H!N4BB!!B!B!B4!B!B!B4B!B!!!!!9'!!!!!!!!!!!!!!!!!!!9!!!!'!!!!!!!9!9!!!!!!!!B!H!B!N!B!H!B!H!B!B!H!B4HB!HHB!9!H!B4H!B!HB!!!B!!'!!B!!!B!!!B!!!!B'H!H!!!!!!B!H!H!!'B!!B9!!!!BHB!BB!BB!B!B!B4B!B!!B!B!B'B!BB!B!B!B!B!!B!B!B!B!!!!H!!!!!!!!!4!!!!!!!!!!!!!!!!!!!!!!'!!!!!!!!!N!B'H!!B!HH!!H!H!H!H!!BB'HB!HH!HB!HB!!!H!!!!!!!!!!!!!!!!B!!!H!!B!H!!!!!!!!!!!!!!!!!!!!!B!!!B!BHB!BBB!BB!B!N!B!BB!B!!B!B!!B!N!B!BB!B4B!B!B!B!!!!!!!!!!!!!!!H!!!!!!!!!!!!!!!!!!!!!!'H!!!!!!!!!!!!!!B!H!B!H!B!H!BH!B!H!H!H!H!B!!B!HB!H!H!B!H!BH!B4H!B!H!B!!!B!!B!!!!!B!!'B!!!BB!!!!B!B!!B!!!B!!B!!B!!BB!B!B!BBBB!!B!B!B!BBHB!BB!B!B!B!H!B!B'B!B!BB!B!B!B!B!B!!!!!!!!!!H!!!!!!!!!!!!!!!!!!H!!!!!!'!!!!!!H!!!!!!B!H!H!B!H!B'H!!HB!H!B!H!B!!N!BHB!H!B!!!!BH'!!H!H'H!!!B!!!!!H!H'!!!!!!B!!!!B!B!!B!B!B!B'B!B!B!!B!!B!B!B!B!B!B!B!!4!!!!!!!!!!!!!!!!!!!H!!!!!!!!!!!!!!!!!!!!H!B!H!BH!B!!B!H!B!H!B!N!94H!B!N!!H!B!H!!!'H!!!B!!4!!!B!!!B!!!B!!!B'!!!!!B!!B!!!!BB!B!B!B!B!B!B!B9!B!9!9!B!B!H!B!B4B4BB!B!BBB!!!!!!!H!!!!'H!!!!!!!!!49'!!!!!!!'94!'!!H!!'!!!!9!H'!!!!!H!B!H!B!H!B!N!H!H!B!H!H!H!B!H!B4H!B!HHB!HHBH!H!H!B!H!B'HHB!!B!H!!!!!B!!'!!!!B!!!!'!!B!!!!'!!B!!!!!!!B!!H!!!!!!'B!!!B'B!B!!B!B!B!!B!B!B!B!B!B!B!B!B!HB!H!B!B!H!!!B!B!B!B!!!!!!!!!!!!!!H!!!!!!!!!!!!!!!!!!!!!!'!!!H4!!H!B!HBH!BH!H!N!B!H!B!H!B!H!B!HBH!B!H!B!!B!H!B!!B!!!!!!B'!!H!!!!!B!!B!B!!BH!B!!!B!!!B4!!B!B!B!B!B!B!B!B!B!B!B!B!B!BB!B!BHB!B!B!B!B!B!!!!!!!!!!'!!!!!!!!!4!!!!!!!!!!!!H!!!!!!'!N!B!HHB'H!B'H!B!H4B!H!B!N!!H!H!HHBH!9!!B!!!'!!B!!!!!!!B!!'!!!!B!!!!!H!B!!!!!B!BB!B!B!N!B!!B!BHB!B!!BB!B!B!B!!!!!!!H!!B!B888888YYu888888Yzyģ888XY__888YutϨáá888Yyuáá888YYáá888Yytáááááááááááááááááááááƅƣġááááááááááááááááááááááááááááá_ţġáááááááááááááááááááááááááƅƤġáááááááááááááááááááááááááááááááƅƤţġááááááááááááááááááááƅġáááááááááƅƣġáááááááááƅġƅƣġ H/ 0|DTimes New Roman0:A 0DComic Sans MSn0:A 0?B DTahomaans MSn0:A 00DWingdings MSn0:A 0@DSymbolgs MSn0:A 0 A.  @n?" dd@  @@``  #1  V .j (HD D/X$2$@9R 55L : 0e0e     A@  A8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|ES"@8*ʚ;n8ʚ; g4cdcd :A 0ppp@ <4!d!dL$ 0ij<4ddddL$ 0ijg4CdCd :A 0p@ pp<4BdBdL$ 0ij80___PPT10 ? %A New Math Type: Tree(Math Tree Continued& Math Definition for Tree of ABase case: If x is an element of A, then compose (x, empty_string) is an element of tree of A. Inductive case: If T1,T2,& ,Tk are elements of tree of A and if x is an element of A, then compose (x, T1, T2,& ,Tk) is an element of tree of A.Z U      ,|STree vs. Binary TreedObvious: binary trees have max 2 children restriction, trees don t Less obvious: there is no empty tree (as opposed to empty binary tree) Smallest tree has size 1 andRoot EntrydO)PΑI)@PicturesCurrent User/SummaryInformation(0UGE      !"#$%&'()*+,-./0123456789:;<=>?@ABCDFHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{}~ DTimes New Roman0:A 0DComic Sans MSn0:A 0?B DTahomaans MSn0:A 00DWingdings MSn0:A 0@DSymbolgs MSn0:A 0 A.  @n?" dd@  @@``  #1  V .j (HD D/X$2$@9R 55L : 0e0e     A@  A8c 8c8c     ?1 d  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~0u0@Ty2 NP'p<'pA)BCD|ES"@8*ʚ;n8ʚ; g4cdcd :A 0ppp@ <4!d!dL$ 0ij<4ddddL$ 0ijg4CdCd :A 0p@ pp<4BdBdL$ 0ij80___PPT10 ? %A New Math Type: Tree(Math Tree Continued& Math Definition for Tree of ABase case: If x is an element of A, then compose (x, empty_string) is an element of tree of A. Inductive case: If T1,T2,& ,Tk are elements of tree of A and if x is an element of A, tPowerPoint Document(DocumentSummaryInformation8hen compose (x, T1, T2,& ,Tk) is an element of tree of A.Z U      ,|STree vs. Binary TreedObvious: binary trees have max 2 children restriction, trees don t Less obvious: there is no empty tree (as opposed to empty binary tree) Smallest tree has size 1 and no children&))Math OperationsRLet T = compose (x, T1, T2, & , Tk) size (T) |T| = |T1|+|T2|+& +|Tk|+1 root (T) = x children (T) = T1, T2, & , Tk Let s = x1, x2, & , xk first (s) = x1 last (s) = xk,%P     P NTree ComponentType Tree_Kernel is modeled by tree of Item Initial Value there exists x: Item (is_initial (x) and self = compose (x, empty_string)) ,T      Tree Continued& rOperations t.Add (pos, subtree) t.Remove (pos, subtree) t.Number_Of_Children () t.Size () t[current] (accessor) 8 g gh Practice OperationZMost operations on Tree have to be recursive Use 5 step process to recursion: 0. State the problem 1. Visualize recursive structure 2. Verify that visualized recursive structure can be leveraged into an implementation 3. Visualize a recursive implementation 4. Write a skeleton for the operation body 5. Refine the skeleton into an operation body0NZ ZN  %Step 0: State the Problem_procedure Display_Tree ( preserves Tree_Of_Integer& t, alters Character_OStream& outs ); /*! requires outs.is_open = true ensures outs.is_open = true and outs.ext_name = #outs.ext_name and outs.contents = #outs.contents * OUTPUT_REP (t) !*/`Z ! 5S,What s OUTPUT_REP (t)?!You Give It a Try! (Step 1: Visualize Recursive Structure &Step 2: Verify That Leveraging WorksAsk yourself: If Display_Tree could get a helper to display the subtrees, could it take advantage of this generous offer? Yes! Once you know how to display the subtrees, you can just display the root followed by the subtrees between  ( and  ) . *Step 3: Visualize Recursive Process$Step 4: Write a Skeleton}procedure_body Display_Tree ( preserves Tree_Of_Integer& t, alters Character_OStream& outs ) { }N~ +T 'Step 5: Refine the Skeleton}procedure_body Display_Tree ( preserves Tree_Of_Integer& t, alters Character_OStream& outs ) { }N~ +T8Step 3 (alternative): Visualize Recursive Process3Step 4 (alternative): Write a Skeleton~procedure_body Display_Tree ( preserves Tree_Of_Integer& t, alters Character_OStream& outs ) { }PZ -S 6Step 5 (alternative): Refine the Skeleton~procedure_body Display_Tree ( preserves Tree_Of_Integer& t, alters Character_OStream& outs ) { }PZ -/p "#$  0(  x  c $ w `  w x  c $8w  " w H  0޽h ? ̙33*  @<&(  <r < S D `   r < S  p   <  fD 8c?"]0,$D  0 " display root and  ( for each of the subtrees of t { remove subtree from t display subtree by calling Display_Tree place subtree back in t } display  ) &0n<ZP-'!H < 0޽h ? 3333%%___PPT10r%+n*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<*<;D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<;A%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<;AD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<;AD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<A_%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<A_D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<A_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<*<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<*<+8+0+< +"4  *"@T(  Tx T c $` `   x T c $4 p   T  f 8c?"`,$D  0 ( object Integer pos; outs << t[current] <<  ( ; while (pos < t.Number_Of_Children ()) { object Tree_Of_Integer subtree; t.Remove (pos, subtree); Display_Tree (subtree, outs); t.Add (pos, subtree); pos++; } outs <<  ) ;f2)H T 0޽h ? 3333..___PPT10x.+n*D$.' = @B D-' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*T8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*T8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T8b%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*T8bD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*T8bD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Tb%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TbD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TbD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*T+8+0+T +r  }K m}$tN%(STahoma WingdingsSymbol321A New Math Type: TreeMath Tree ContinuedMath Definition for Tree of ATree vs. Binary TreeMath OperationsTree ComponentTree ContinuedPractice Operation&Step 0: State the ProblemWhats OUTPUT_REP (t)?You Give It a Try!)Step 1: Visualize Recursive Structure'Step 2: Verify That Leveraging Works+Step 3: Visualize Recursive Process%Step 4: Write a Skeleton(Step 5: Refine the Skeleton9Step 3 (alternative): Visualize Recursive Process4Step 4 (alternative): Write a Skeleton7Step 5 (alternative): Refine the Skeleton  Fonts UsedDesign Template Slide Titles no children&))Math OperationsRLet T = compose (x, T1, T2, & , Tk) size (T) |T| = |T1|+|T2|+& +|Tk|+1 root (T) = x children (T) = T1, T2, & , Tk Let s = x1, x2, & , xk first (s) = x1 last (s) = xk,%P     P NTree ComponentType Tree_Kernel is modeled by tree of Item Initial Value there exists x: Item (is_initial (x) and self = compose (x, empty_string)) ,T      Tree Continued& rOperations t.Add (pos, subtree) t.Remove (pos, subtree) t.Number_Of_Children () t.Size () t[current] (accessor) 8 g gh Practice OperationZMost operations on Tree have to be recursive Use 5 step process to recursion: 0. State the problem 1. Visualize recursive structure 2. Verify that visualized recursive structure can be leveraged into an implementation 3. Visualize a recursive implementation 4. Write a skeleton for the operation body 5. Refine the skeleton into an operation body0NZ ZN  %Step 0: State the Problem_procedure Display_Tree ( preserves Tree_Of_Integer& t, alters Character_OStream& outs ); /*! requires outs.is_open = true ensures outs.is_open = true and outs.ext_name = #outs.ext_name and outs.contents = #outs.contents * OUTPUT_REP (t) !*/`Z ! 5S,What s OUTPUT_REP (t)?!You Give It a Try! (Step 1: Visualize Recursive Structure &Step 2: Verify That Leveraging WorksAsk yourself: If Display_Tree could get a helper to display the subtrees, could it take advantage of this generous offer? Yes! Once you know how to display the subtrees, you can just display the root followed by the subtrees between  ( and  ) . *Step 3: Visualize Recursive Process$Step 4: Write a Skeleton}procedure_body Display_Tree ( preserves Tree_Of_Integer& t, alters Character_OStream& outs ) { }N~ +T 'Step 5: Refine the Skeleton}procedure_body Display_Tree ( preserves Tree_Of_Integer& t, alters Character_OStream& outs ) { }N~ +T8Step 3 (alternative): Visualize Recursive Process3Step 4 (alternative): Write a Skeletonprocedure_body Display_Tree ( preserves Tree_Of_Integer& t, alters Character_OStream& outs ) { }PZ -T 6Step 5 (alternative): Refine the Skeletonprocedure_body Display_Tree ( preserves Tree_Of_Integer& t, alters Character_OStream& outs ) { }PZ -/p "#$(4  0(pt(  tx t c $C `   x t c $D p   t  fF 8c?"H,$D  0  display root and  ( for each of the subtrees of t { remove subtree from t display subtree by calling Display_Tree place subtree in tmp } display  ) swap roots of t and tmp // MUST DO THIS!! swap t and tmp&0n<Zb-'6&H t 0޽h ? 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