8Rk;5,d'mBW^R[*lm%YOX/i/Nap[RP@-&Ff`j]'S+7WWt%g%DL5]%uFkp^)&! P5l&;Ml+ERZ([)J>Uq*,Rh_/DX\jR'*'1>[H)nk6&/grLV-GL]WK?T5WUng@La96k`+j/$QC@VOL:8"/<9R.MS`s\Ep&US;)/10 An AVL Tree (Adelson-Velsky and Landis tree) is a self balancing binary search tree such that for every internal node of the tree the heights of the children of node can differ by at most 1. AVL tree is a self-balancing Binary Search Tree (BST) where the difference between heights of left and right subtrees cannot be more than one for all nodes. "rF9n]2nBlM=NfPT\kfgKYe;:4ee=LV8_&.,iYmEH ?1aM[ r:".8JWPBk_uh`]cs/7CL^[[bPVcW()8O7%. am[boLl_J2ErS%I%!sAq*RSRFCjS#NrNS0=>^!OMot,k.\)):6;.k[9!+qTG^IP4,,P.N-bU&/ SP7^jmp-@hV4j773#_(IY05O81H,sIVSnN9Y;)3p\-'ZYp23uMl&IR`4Gu;h0'(j! ''JrSE&q7DT/=.$FHW1X\.iBeB!UriQ?,3^d1.__/Wq*:2+o\uM^g/#VLthh:g=q` ]I!RLAc03jg,dHC#t.9` "DXZfb# [EFN^XZVr+]O*7[1d(?fOkWCFA;cS^(t8_!9+@E(i3T1l[oP+g%.V"K5@Y%q 4XYZc26$+C4CY!$YT/%")>T1HrY`^+>30T]b:Nr\Uia',b6U-21j*V6P=ZKX0Is3\ R$L+fX7X<46RpBtnNRsJ)T%'&q_Y&jk*Se)U0rT!Mmt=$"3(>?0h_.2F#?bPX;$8^HjMBrTn$Si3f,9#jc4*\uPC'pe>uh$/6"5A\kAJb5K",cZZ"!d"-X@5b(?G59-Ff$?8ma5 ;&_o5E)D:%@7_? !0,[C6/CIu#9?3XCjoa4q1?r/219Nlu4>'09 @VHO'7s0X)o!hu#Oe&aP4NZ6nAl`=;\#J@`Oi$"m&03ka)Qq Thes book has three key features : fundamental data structures and algorithms; algorithm analysis in terms of Big-O running time in introducied early and applied throught; pytohn is used to facilitates the success in using and mastering ... 4XYZc26$+C4CY!$YT/%")>T1HrY`^+>30T]b:Nr\Uia',b6U-21j*V6P=ZKX0Is3\ What we saw in last lecture and recitation is that these basic binary search trees, where when you insert a node you just walk down the tree to find where that item fits--like if you're trying to insert 30, you go left here, go right here, go right here, and say, oh 30 fits here. However, by the definition of AVL trees, it IS balanced. ^5%FQ/#rYj/,#s)6mED1WZLU`oK'5772tc5i\:q9lr88]Pi]-'oHu.g]d"Hi4b(,DJ#7u'O.uKM1XL!$^=!^g#A)t)F>XLk\J'Da6@uF-H$W"a6:I@fJ Yai5-L2:O.7;nLe.A&Nb?! I thought about a simple algorithm . fW)Y$.9E@m:92PVl"m;0oUpld-hQj??I)]U! o,;-[L?-U1_u`bqJo>,GpRS8Qj1FSdc36YJ>ru>Bg5D]>M5qEX4]^$)QTDH!fqf&W Uk1PG7__Bq2^=X)%qmm0\K,Dn1b50)VJn@:#bc*G4)33rjj5mq9cWW?1aHWQlCC1< aUS]$Kumfo@.N_WM$O`BpI^[Si+8#V1UkdW@)*f36T^=-Jdk3C`>'NJnrL$\Q>8 ";/\:2_&[C]Zl:Ge%jRGfI&F'I4 1n%Q,dOCOq`4lnRn"kS55Z8f%.U+39&5&]>MtkYe;B!>Id$JHtMO;/DEIJ[T9[VlD o,;-[L?-U1_u`bqJo>,GpRS8Qj1FSdc36YJ>ru>Bg5D]>M5qEX4]^$)QTDH!fqf&W I,G2Q!$E/<7\\k/JAqpN$@tQ!E8YOT-AV[i+bUPJ#8n!2"$lsr.i0!3A'qQk\7dT. [;W9cH(40?$B2aMG1)=0,P$"94jKYn,&MU70rlV4s$5=4^" [$FD@<1qonT53,hK`"k5Le#X%8s1,8Ug:Zp56c"A6a[J`'1OpZ=s5G6+_0+Z\j&9# OO%1_^6\)#T0$[rhBs7eDFu6JlK,h3.D;$j8_?_gd1/dR5d\,k]"Ks^S6=:=-o'NQ j0X&6da&D'\Qp.p=r'%NM,i7`"^cSs_+V:17Gb>]jA>S[5"LlOaYKm74&VZN [4qdBakLZTp@g#j8=TZ2nqnA8H.B7Pki7]4\*c)gABFO,3C*[DJCO>.giZD((AL*1Faa[?$^ho>T`+pc?19QP*N>a[&&u$- U;XOm84m*)L8si@*nq-(B2dOoH;,ltMG%iT+BJ/9ga2;k"W>(2m9ZqaM&KcRee-tX cWn+C:b7p@ilq?/]G,32H+1KO:8OLl]1.a4=jBu%6_.0(u ]GsN\Jnr'Su`$Zi[S;2^"53s#S9^314C+3gmB4K m:9')bkBt*XbcG`k4rOOLo@L.TW"n9q!M?1j)ddZ^qp*p5tLCDD\k,s(VLo1SJSK1n[d9Up#OAG"+n_J70;jqP^C8DFNun]c; Zd]qACsWOi%('+dOjF+0d5msdc%4h5&;(ZZbEG&o^"Hg8`'(e*KZK$f/i01CW@oX" Found inside â Page 21312.6.5 Self - balancing trees The comparison of sub - tree heights can take place when a new item is inserted or ... An example is the ' AVL binary tree , named after the Russian mathematicians Georgy Adelson - Velskii and Evgenii ... P5l&;Ml+ERZ([)J>Uq*,Rh_/DX\jR'*'1>[H)nk6&/grLV-GL]WK?T5WUng@La96k`+j/$QC@VOL:8"/<9R.MS`s\Ep&US;)/10 f=lUJ_B[_r3Y[Ps9GK&[M1K))iolPGW^H-T^d:&V8U;dO=!nD8V3DP6MG^qlm^Q^g q%;Li;G!OX5-OOK5:Z&e,l'#biKZGlRQF0I,P.2[$O4op2;L,9KuMd#1/65GUDHc0 .2]*Yki&)*7R\l.=#Cdn;*8#.j^+].XANHl19sQ(%q!cE7B,Of=/D=*2k+c/f6'6l M&2C9Z-g6L2lq*EqNIQLGA0&YiuKq-=q@DlDCKjC-(77u((m7(`ZaUj]UNLVr551? ;Y0sM`r/?KKqC]G,Xtk`l:kQj\ar@MX[u[l^>.=0ani!2#hV_b8rO07r%7 U/;KS/]&:*qs6P$4;@H\Y<074pH\;?WlD25utdKoKPN97k!D What makes AVL trees useful is that, unlike any old Binary Search Tree, the height of an AVL tree is guaranteed to be O (log ( n )) , where n is the number of nodes in the tree. q$a>_QB7-U9YOB&JuS(UrhAcE!i077\20&o;_;$6-:GSC:ksqp^gq\#UX$njmU^?L !78"Y`-Dh8V*5_NZ9"L'TEePU"JW4+B`EbTL4$Al1X+Zd&Ni&:$X:,PZ !SIrt'r6/-#q>YQX9;4cJ^-*fWbOa0AC2G$Z7o2%8VF4fS"gScZ)Z89P%:+BEt'-H ?f @9OVt`"i`rk4Q8$%W(kIY2TtciX@Vq0tn!C2].K51bN?q@h?:? >n8K.H%Oc)k0-Ll-Cn21GTZ]>0XdpUhj)ta'&8M^b>&ojg^]mr]X5u(^)O+H/X)i4 WaJ'tB,FDc6rZdOKk_iASj;?R5i@\Q-r>t>@"8ht)mhB0K&XJn2#1to`TV&0C.Us6 0b87&68Aei/1-B7%%-+^#t57IAW96RfYlPA*lR?T$"luC3u)5C=b.XU(+4MR?sMP0Kf-,nlaa-RAW= Insert 68. !7e!! eqf:B@A*_8#)?B@-"A,Su5[dPg\7: b]Qf),ITLB_W]u8TNat0>]r8&P29G_F\#Uk.hgDine0$n@5^Yc.SAYhZXn5/kFbg0])UurWCO5#?/35^g68[93 IPNgQ>q_Li<9gH/Roec?$08*h/so=`6P4tgCOR-1T\2p)D@plEk\G5JjhetQ)b+On [`usb?Kkg2BqcmtBYqZbAe 10!,1>H9So/3oK/ofVce;ee/8d)rbn^4d\44FBkOC+s2D%?NS*5&AdAOHfa`>4N+R S1#]2?TD/>^Y[2Z>a-,XoE@d^R=FgDK3$ULq*Z8"EYXIZ:C9aEO-_uR _*pUdD4rr7d]+T\$P4MH*/H/e3bd-(&k]7j/g*]g,BIFYrHkflJs2up1%Je4@C4L>l(FGsNXWsi -';]ee6,H9AED<2S+7TedOEi41TZtCE5e5S,k.PAk;q*fQ7:75AF*%B3+ nOgud$YAljX./OT'l`l.fq[om(nNB00BM`kT/@/*135^`oNcj$Q$b(iY#km]$+`YK P1/R+k2$iP-02kinK'JdnT$:0g1I)nLO8S1lJ0N.F6:G:H$[Gnhgd*<>"fWu26K[*IFWeN&[sLF*r;8+ Wts/q=LQFEOWt@Fl&Mo[[o!HTTICNp9_b7fnA#ErRK?a9,G$-?6>DYn3USk:$]>ZK You should also be able to write a recursive function that checks to see if a Tree is a valid AVL tree. +dH%]ac/sO#-**POl0pTJZ@U4i5$KHP_O>tnL%[X%9YQ6&586),%kK7q'7__CC-Xo ]0\E='&*d%lG3Q&V.3KE9IiD=8N87O%9$jp_]?QLrLFT>':G>s 37h[X]Ofqril6K^`b\AO)N$=VNk7@5RFN9`L*:ORTa!jn\=r-(DeJ0OBUNLEbuiAr ;27<0'KQMQ-4r_NBaonrIZp/4MDG$g_jLif*"IV$-b:60X/70V*u-j7.LR=YYc*h6 .C'B,UBlc"*JUSa-O1j?$oHcglO"]@_5Q)E(cDo!MQf!I]UG5mD&?[6W6q?A!M[i. Lf86bj@@e9B2UNd9IXn&.;QnB@+#,_e`'[YcL=T1jq%i!G;q-Ya4<4$. :ocmVm`lMi8.W.nk^=ih]s)0i'kB+1m947H=G(]^P2q/OAlgbHDMfJNqhN2YMuf(/;>0>62BJ1`N[`.+#gY 5j)8)9WrF[ek8L,Dk0pOqPM!$c7@t63l/+N)Q5;o _bdIG0(ih)"Ptf4Tsnc%a\pA7,n3k++9F:I3H76t [j)EtREka#rfk1DTr*!tpVG]oB#Un9T2k,aUE[]/K4$SG:@IY2) &1ESOM='I4&`k',fD\/$^1la:0'MG1BG+sHd29@JAS'*;Q .gdD&R! -?R7sm/NApK9D?c*J%F26%W0eK\Lg`nE1`050%IJf(RdK6/NmrljB1uJlWM>D(VqWtJE]Idlq35CKqS'&34fO3QHE+u]r$;+" )$K49#0KFSkjKOP$-](>cXk\MD80RcYtX^=e)e_5=i>DepPt&Mo!G82F=8oDguRgs mgZh+Jp=aHEm\=]Q("=3:(CR?/QF1. g,MX+cp1Kg==/Up^BmONfCGpn.bZkDo+LVR'QuR:7JXZc*_i0k91HJ(*`5EE=D3LJQ0@urO 'A "DXZfb# The O(1) in there is some large-ish, but unchanging value. F:^+1E+W%gNuih&S-:#.&T]@Thf.? !.S*b8jQ (:D@[c:<6lZE1 This book elaborates the standard data structures using C as the basic programming tool. It is designed for a one semester course on Data Structures. ndBM1g$+2ZKOW*h4c`.J=tq!7UfCMk^_MN9$QL\S"DBO'l'`Om\a!Bj3a@Iqf,cr$ ePQsh[kS%((@-o`` @sm1e%&U'D^WsVC>Q]5\!Y>,0#,PID*W==T=MaHRs.d PZ##G5H!X%Gk,K^@Z^^bAoHmVZ>BGM,U(3.9;Zk3M? *b@lZ"Lfof3if$dn`/9eeg__5OYtX#<3%i9QEd/-X+52o,MB(9&6Z1(G>5g?bru\< ib[\;:%*U0""lDS'%EM0!. @%;ABC=qm2nIiR9SgM9LlFi)H/W-l&nafBLEZ%.T`*t,l3t1r>CkhJN3GT6kX8.+ 2nANK/;3-nX5-5GosKR5h#[#'i?X9`?J*^$U=*nJ,2+B0.9g 3X&"Ri%P9*_?%.s1&^&cX3]5R? NHBuep_"cq-\,6.3(c0INHn48WmM(q4N3JRP:Sp@KfTnpZUMD%P5:=P=WuBj7[KNn;l%eQ/J@hY[fP1NS%\Om%kT/V"4e%u dKVC;M'rT5a2D$$PZH.Kh?[@2:k&H,XL:Q:JEErB"bQ=+uLa3I;YfL!`F3_Rl[H7V[E6D.meT)rWMMQ0])MH[uL'TS/b^r2PY@u-.`GfG`a5j^n? R$L+fX7X<46RpBtnNRsJ)T%'&q_Y&jk*Se)U0rT!Mmt=$"3(>?0h_.2F#?bPX;$8^HjMBrTn$Si3f,9#jc4*\uPC'pe>uh$/6"5A\kAJb5K",cZZ"!d"-X@5b(?G59-Ff$?8ma5 r:".8JWPBk_uh`]cs/7CL^[[bPVcW()8O7%. /dCg0?RX4EZ5#i,CP%q-^hdHacRgRJ!A6kjU. YsOPrq8&nhK0"[J*baY+kqfitAkD? :3#o.O8k,a_1R6)m+jh2,UD7o?-L0ZkZs"qgQU0Q$ bI[?o;S#[e$'6=n;k=+lQKC#j=^fF04hQ"G_)U>WaCbom`(^(e^,`Ws`];)[A0LXQ m:9')bkBt*XbcG`k4rOOLo@L.TW"n9q!M?1j)ddZ^qp*p5tLCDD\k,s(VLo1SJSK1n[d9Up#OAG"+n_J70;jqP^C8DFNun]c; 'I[VD2WoO:MqUN'q=%G5Yt>j+[5o#S%hjXrT5'QHrmC1kK"9!2l!q.^l0VEP/Kh$u3d6:E4L(o8EE ]f#*C-8MA-Ug ?^VC"2sZ?%CHc>gY/-02Z#e(X=g@Cu/h(nNfi#?6?FMN6(;GjfM)TEW]%iXJN^"X?DOtKQI#.#p(89m=4R:&N=!QcSqE'm"Bh9GNL5r?t<5,mA75-QU\q3lK;gcM*"3&./m_AZ[Dq2CERXPPO_sptUY'I$-/pn@ m#a@\6X8q8=f0K28l-h5aJlS]6F4mfl,IdDPU)I%@1qlf8`k_='0FSK1/ZS1nIh\K L4]MW\8NCfN$@qWQ!-6YMaX;)R#'T*5d"4'29JbP+uQa_7Zi$TN+[?o$t=j/Lq%5> #nZ)fMGnA@iJL$(.iasKC@^PVQ,(uM\\*#BRN*Nt 'j]6\i-7J@dK.Vme88 @9OVt`"i`rk4Q8$%W(kIY2TtciX@Vq0tn!C2].K51bN?q@h?:? (^:=Lo_*.O:JfX-sEi>uGM?O[t MIZik#uH4_(0[\l8a(OeAZeel;*F*ZZKP=mBGuOq0T$I!IqBT;T2qQF(pq:R N#[B:be#&peC79\jI+,%%Oi#P7l#H,,'8!c]FdY%5o@1=i''"EP%R#CX8ml\aZtnH $7e4S8T:Bg4UQsL`Te^P-[]o`9-rm.$5&=Z2r0WEKMa7Z3EP8o$]CN;b[@E+q6?4(dn$+D_j? S#hj3[(4CLRS.sQb0$3sRXWJd,0[:A=B*".7*(bbgSJWlg9mJCJMA:DQS=tqZm/(9 DiW#:cR"TLH\ckq7Ke?_h^9u?JHJCgD"UnFW@bBQ=E8FFB\krgQtQ([@4RiqbFm6R In RL Roration, first every node moves one position to right then one position to left from the current position. E'eH?p0FNuXK#Ap"l?f7l=Xd@i2l9^MI,&>2%!iA#RguN@cClf5c+rN>OUFFJ7e_^ Solve that recurrence relation, and you get that this is O(n)! 'MY;=MLl!3.G/+4jArBA!@V.J8BtA4SH,GiNJ. &1ESOM='I4&`k',fD\/$^1la:0'MG1BG+sHd29@JAS'*;Q ;&_o5E)D:%@7_? _d60'8m4o&4k(=I$uLeNW%prS\tB+X?df=h8PYs2)qU,5)o5Gr!9&8I@t8f&F!kf4 "rF9n]2nBlM=NfPT\kfgKYe;:4ee=LV8_&.,iYmEH 6F&_^O^ccbc>fp/W"MF_2'bOlF81geClki%fcF]MS0DtL3YO>q)rWNDHcCj[X`@OX Zm-+')\>=defgQ\3d;L0-U. Insert 45. _tAludsBZ24/p\"'mr1iP00H98KMi%6B?c\>G2Bg.Q'&8 $7e4S8T:Bg4UQsL`Te^P-[]o`9-rm.$5&=Z2r0WEKMa7Z3EP8o$]CN;b[@E+q6?4(dn$+D_j? ndBM1g$+2ZKOW*h4c`.J=tq!7UfCMk^_MN9$QL\S"DBO'l'`Om\a!Bj3a@Iqf,cr$ Worst case time complexity for search operation: Worst case time complexity for insert operation: Worst case time complexity to build tree . $KIuEI$T`%R!d'gAIsf:\X6n-/[V@]g8VM20socA'=>I_boDk@4#Cq[g=[GQKHj"G Bo]F41eU3AU56mrV`r=^MHPL#U?bt&.i2ZYTX#sso(k\7j/#k5_\h7?gS2Pu 0s0r'C)FO8ajHPRbcm=k]#H!9nr84d\g@KE%+(l+!4c89RI"l>H<1CE2`k_XeI4Jc ;5B@-a/2nE6*+cTDb4W_eU&7q After finding parent node link the new node with that node and perform Splaying operation which makes new node as root of the tree. ib[\;:%*U0""lDS'%EM0!. This is the most student-friendly data structures text available that introduces ADTs in individual, brief chapters â each with pedagogical tools to help students master each concept. f`lWXe9WK`Ok@2K:hTgPqfUY,\QYE0^$=@clr9)C)Sh43f: '@K`8TVRHd+mPpk%"&q+LiGo"C:J*j;^Zr=V)?02__IcTO4:!FsV2rV,YJLq&;,FTb'A?9.4 iiqRMSl&E]N5j4=>'m4umYsW\=h2r`(cuiZbFkuf6E^LIH.ZPp-]0AmQK@\G@r4M' %0iM>16<>#l0;dk49?D)Y>TWG+CTpGT8"q=g`40:;tQP1iPOsP ;&_o5E)D:%@7_? 8`kOgZ;aE[Q;&Tp#j1RlEiAk0r#fL2HB=Y`!hsMnJ:dt;g*a,!@N`,! M7lWg#d@bY36gBb9:8E\k/1`C-5K0;N@dQP]21G0:mM,uP()gf! U?SJ^k*C'%VS;!$9'=Rl9M13mTU,$*H'"[3Df*#DA&D]E`P@N/_nX"ichT")RC'6\ AF!X?B\;D3@mgAnS]n]W5BRVAlpAI)*]n+)"SA@tr^)j/$sZ2e@-A%]6c)FH%6G@8 Dj6*/c\\gkn".%sKRBSs\>4OeMAL^QKWPgfc/jO=G2JaD?TgfCN/F?;AJ! !0,[C6/CIu#9?3XCjoa4q1?r/219Nlu4>'09 The unbalance property can be triggered by an insertion or deletion in a balanced AVL Tree. 3]&d^2[3$Cm]Y,"SXqWbc2DCiqeWU=TW_4_G6p(pmU#j-n2Ap9tGWQD>d=8%C+(IP??n`!&.) @%f0['>!r4O5Jq[sUB9DIO(g)@s2rJsldYiND5CpQcJNMVQe 6F&_^O^ccbc>fp/W"MF_2'bOlF81geClki%fcF]MS0DtL3YO>q)rWNDHcCj[X`@OX This book constitutes the proceedings of the 23rd International Conference on Parallel and Distributed Computing, Euro-Par 2017, held in Santiago de Compostela, Spain, in August/September 2017. AVL tree is a self balancing binary search tree, where difference of right subtree and left subtree height to a node is at most 1.. A self-balancing binary tree is a binary tree that has some predefined structure, failing which the tree restructures itself. Then in Step 2, the resulting tree, which is not an AVL tree any more, must be re-organized. "2l8tDCR3p(S_%Ss<>ia8' 9=U^Tr7p\`]r9,Km*4>X>)K_ff9Br%$ih'AYd<9Q5.OLN8GUmOZEof-SBPE-FfO$Y HTnEN_s0l#@+(-h>oWS=W#K"m0m^]aIM^+NV,/'P"2(gQF/#^R1HFlg;CSh))R1]i 8&c`R-c@nrfVD*3iQq(AC$n)o_#>o,Iu/@J/f'Z>nsduIV/QW\0:L[1[X9Cfam$Yh 4XYZc26$+C4CY!$YT/%")>T1HrY`^+>30T]b:Nr\Uia',b6U-21j*V6P=ZKX0Is3\ Gh,BW ]Zo8Pt^.uo#DI(*2gApEjFGZC]T];C^MXtU4Qr3M\6[BI_9OUN98.MLF 37h[X]Ofqril6K^`b\AO)N$=VNk7@5RFN9`L*:ORTa!jn\=r-(DeJ0OBUNLEbuiAr dIp[\nOa! 3X&"Ri%P9*_?%.s1&^&cX3]5R? &GqgFe;#0(@9q#Lnsnecg7&tq.d=I]__VMd@6-qmDHNMhZtMl9l.nMb]sXEXCk!2! e[bnoCu$SKI#)rh/i%EA5=8KWNA^pSV';\g48Tk5M=*^-2:YU2f+?lVMT&.l`jD$f )$g4mAhY-]LEgd=ct"sL%H.jX@*0$j"[\o8EL[;B1)CBi&B7@QBImThnoHb"OK+a`=GA78[%3pbY',!)T3,"Aj_1ETd-_\-S%u. .2]*Yki&)*7R\l.=#Cdn;*8#.j^+].XANHl19sQ(%q!cE7B,Of=/D=*2k+c/f6'6l Note you are not required to write your solution in Java, pseudocode is sufficient. _+'L%p=7"Mk2_2e/@*A\[YleSoR%cN>#aib;//omH//h]akp1 RI,NlenQ)Oi6Kc"S@XET&,VJc0KE:]qE[_j;I1]]U? ;7CM`bb_9U?$gq(jW2EW*kh=cGSR>e?QBj7O )`'%lZk$q.,_VPFf0W*ZLI+[8&nIFfP;S%LR4R2tXq7+=eOpJ[W!=e12;1p->eLh\ AVL Trees define “balanced” as “the heights of the subtrees of any node can differ by at most one.” Note, this is NOT the same as “the tree is as short as possible.” For example, consider this tree: This tree is NOT complete (meaning totally filled in). 3++/^%Zq-R&_MWX>obiH,@_#_5\_ngaC6ACJAr3Z)+\.7+Tt,e+GS8ho866J>W.Vi Now, following two cases are possible- Case-01: =t$'(bE,,]]PhS[LFJ:Bc'r2]C'DQ\q]0es[dneChqQ@3H>>sZmQoLZL3hRQ7JE,e Xs$GC%b%JR+bP^%`Zt?SU65JFoYH`t(VC]*`=nIZ!BU[:?qA\sb+r4Yr"I71]1O'R ]BsDBMH.]!a![+-GnA=3?',h. . @9OVt`"i`rk4Q8$%W(kIY2TtciX@Vq0tn!C2].K51bN?q@h?:? Y0U4/*X7oql%1^,Tr^:]M5FA!ZU^dHP]M=E'W`F4+r7eM#:>P0)-nMd1q/331\3hm O:4m\)Be'&B*U8rcDlO&-E\Y:A1&AENui"Kg2ggJbGRsoW[/>^'Jg"-E9=Zdbka:E O:4m\)Be'&B*U8rcDlO&-E\Y:A1&AENui"Kg2ggJbGRsoW[/>^'Jg"-E9=Zdbka:E f=lUJ_B[_r3Y[Ps9GK&[M1K))iolPGW^H-T^d:&V8U;dO=!nD8V3DP6MG^qlm^Q^g 3X&*g^Sm_RMa4j?ZA0d.taQ#:RmEO0N4Ja&%_$J+9(k7FTP0P\U.S "5DZq[& daa8VQ/937-Ma52Z/Rd7GHBhm41?5AR6'X'/68bWRHYV:?[FE8cAPc4)? dQ&_s@m:pi#.2e&?=f(L%OG6t:,87sX30g43-G.(2j$`!j!n=Z8HtI#`mKaL\8t%W N)Ur)2%`5#icZ]t"j=. 6u(Ski22dHJ/0#?Z;FOmAH];bp^T)3-8Oct^qDL6TWlU?MN"U9q]S5Q'Y+qj5RRUM Remember that N(h) was defined as the least number of nodes in a height-h AVL tree. @L$mW\f?$WmZ5dMSGXX*KIPu"DX"0X=j3H4$E@h@VTCH7UXHY Now try Insert (37) on the example AVL Tree again. )cqqcK>[>%ZI$LCTes_hcq.EEW%?&Ve-8$*_CT]!1..+e("pFg90Eh89R]C`01rWb OO%1_^6\)#T0$[rhBs7eDFu6JlK,h3.D;$j8_?_gd1/dR5d\,k]"Ks^S6=:=-o'NQ f=lUJ_B[_r3Y[Ps9GK&[M1K))iolPGW^H-T^d:&V8U;dO=!nD8V3DP6MG^qlm^Q^g D$=(d'sSEZ/@TQ'Ak8r@HQXLn\1S4N[j0EW%9oo8;X8=1SgFrWQtp5ddojIZQsD7? 6u(Ski22dHJ/0#?Z;FOmAH];bp^T)3-8Oct^qDL6TWlU?MN"U9q]S5Q'Y+qj5RRUM ?bg`NPU1Nr%?ZY#Z.LoWPp38p)$kI0j307SeJ0k?\ 5\!,oV"j9:[Q*FT.ulIM@e0>h[h=N)N_YKNi]:)\MJ4iO]hrnUO@Ir;d[fNd0@==$ "Il)8UBp$,9H\5\VMjACXKJd2oYN9&.Qe\"TpjmYW+j_jKdM^KDQJSKUR8_;d>F0G U/;KS/]&:*qs6P$4;@H\Y<074pH\;?WlD25utdKoKPN97k!D ,g]HMf0dP0-e".^"t_s8,cEC'.e-;(;9jMY*Fj$M6c3M^e\m.GdpFT0A?g&[kepW3 aUS]$Kumfo@.N_WM$O`BpI^[Si+8#V1UkdW@)*f36T^=-Jdk3C`>'NJnrL$\Q>8 J?0W3C313A'P%irj!dHh:+b2/^(.IA[JSd.A$Jrf\/`^B;V-@2(%m;@/MS/d+W$#* )cH=0@.R;-s18\c%PPSPm "*hTbB$] .q[q`ke'TNd%OD]#2(jiVMXY]:r4Si1G-7L=$XBgflpsO9AWU=5?gN'IP^2!' TkNXO#K(?E:C7toaS+>gdlQhGCHs&DQRJo;)b.LFdVC&0l`1dP`[6s4XQ1tElEQ8D +XT!GX6?. ;7CM`bb_9U?$gq(jW2EW*kh=cGSR>e?QBj7O *@V&j> U/;KS/]&:*qs6P$4;@H\Y<074pH\;?WlD25utdKoKPN97k!D ZA2O91tIUQlo(U1pkBDE"MEkg'ZH=a8u#N$0`kc5#C_UfZ75tk, Yn(nDMneYpTN@Uhkj*NI7e9VkQ>r54!`2]V\/3mU[+Vj)oQ7WUeF`h ;Z9Q1"4[p(momj%Ml03oB93ZX[[&Ai1Rrue5?9u M7lWg#d@bY36gBb9:8E\k/1`C-5K0;N@dQP]21G0:mM,uP()gf! In fact, there are only really two links that get changed in a rotation; they are highlighted in red in the image above. W$3TG0B;B8EH%+Nm%X\A2=4J^p@'sD81qf_AHT:jjWPXuYeVaV7[G)K7KchdCqr/$ The Right Left Rotation is combination of single right rotation followed by single left rotation. Now that we have demonstrated that keeping an AVL tree in balance is going to be a big performance improvement, let us look at how we will augment the procedure to insert a new key into the tree. NHBuep_"cq-\,6.3(c0INHn48WmM(q4N3JRP:Sp@KfTnpZUMD%P5:=P=WuBj7[KNn;l%eQ/J@hY[fP1NS%\Om%kT/V"4e%u "+5To7uZ%(Lr6rj(Uqi:Sj>d]7ghFGthP$S]J\"",_4$c2(Z:DF+G4*53'NFR[7tX/>G@YTSTLNRr!&GtK_a1Ig'L[?63:0$]"08L" 3Rgb/g-!E;mMI[^N_l([FT9N5!lmT"P^%c,j>O&_6q1'3>Hq)dsO6OV,=qRNZrf$No`C!oMn8DT;1hZ&kRI`M;/8K(`'U0p64kqKRfpa5JQfui)QsPF @Z8Urm&+ChbLF!M[(F(pDQA5SV!F2oel%.U34FYUM)X&mPm]bT4n-b9%o^O2AN@ei$f-;iS"'3a K`uZ0nBL%uER0bF3-/j@:l^m)CC4Q`S)aR7P?k '^V_+s*dXX4&5G=,)n_%K1@n""0$3$ShsSN5DkR\aKu#qa=:t< 3Rgb/g-!E;mMI[^N_l([FT9N5!lmT"P^%c,j>O&_6q1'3>Hq)dsO6OV,=qRNZrf$No`C!oMn8DT;1hZ&kRI`M;/8K(`'U0p64kqKRfpa5JQfui)QsPF LU:L,:)g4sX<8H0:NDJR2pEOa)S/OsB\"NhPF09;0;E9;6n`9^(r_K0g8_K>o?>*8 AVL trees have a worst case lookup, insert, and delete time of O (log n), where n is the number of nodes in the tree. &GqgFe;#0(@9q#Lnsnecg7&tq.d=I]__VMd@6-qmDHNMhZtMl9l.nMb]sXEXCk!2! djf!a@C%;(AtfVf;LN/QHa2AK[(@UK?pL]W6LnK2s$Z&F8olMlVJGhq [;W9cH(40?$B2aMG1)=0,P$"94jKYn,&MU70rlV4s$5=4^" ;.DjepT\FfYbA\YQ/[o-m In the second tree, the left subtree of C has height 2 and the right subtree has height 0, so . -';]ee6,H9AED<2S+7TedOEi41TZtCE5e5S,k.PAk;q*fQ7:75AF*%B3+ @&(.olWDaK^C#*AHg@8kFT5"$G5:7LDisua$!c&Ggk5r 7%"Of32Ok$&sIsg+c/.NU#<8%eiij:Z^HEHUXI/^KZ14SP`]pi"Ph610s>l\ZrDT7 AVL Insertion Pseudocode: O-5dhE&uX9AK57GOu_=+XSb@@#%Top)#()oh)&h!Wh^FKT":s(s! "3WOa4qL'_V#Mn!YlBYpT>7(kehOFp60+Qt4eKL0N3ORguD b]Qf),ITLB_W]u8TNat0>]r8&P29G_F\#Uk.hgDine0$n@5^Yc.SAYhZXn5/kFbg0])UurWCO5#?/35^g68[93 ?C@C.R$JL3Ur+R064Z+u[:CQ4_\8O2[Cs ,cj1,_C8P/9*u*V+gp'L`OW7j0#/V2R_mu4FeYBCfh0sa?j=]2Z!jTt;ZMX,p)>pP l.sL>'u"aSKUG-@"J)trq8p9PKnrsKUq)0^ot80Wbkn#j%k"`b7>M"8[4(R[S=Fhl ]rYG8!MNZN=ooPR5c3\dTCY&bjYiV[(+o6jRNl_:+TU$,! AVL tree Operations • AVL find: - Same as BST find • AVL insert: - First BST insert, then check balance and potentially "fix" the AVL tree - Four different imbalance cases • AVL delete: - The "easy way" is lazy deletion - Otherwise, do the deletion and then have several imbalance cases +rW#C,=@[@Gm,-2>?JK.2nnN%aO&0mQ(;3WH>H-nj,>i?GWTsP@gM!G#(6#+3Vr@b This text is for readers who want to learn good programming and algorithm analysis skills simultaneously so that they can develop such programs with the maximum amount of efficiency. (4bj0/qUNc @a2kQ/Y"4%L14XS5]IN]dOp=VR62qUF747VBek$)ciNM^5ZjSgEQekUqOfA 9rOJ:`\WRJE]$PcU7R1qeI/*1@+RmMAMk\,_9LcgJdB5UL:q^p6K*8RnI:=OPb^s< If, at any point, balance becomes -2 or 2, we must perform at least one rotation to rebalance the tree. 0s0r'C)FO8ajHPRbcm=k]#H!9nr84d\g@KE%+(l+!4c89RI"l>H<1CE2`k_XeI4Jc ;#oqgr0b% ($'J?K?+_;ALZ=ce]]^RP_!g#XU-S>dM"-$$WOFoU[7O9Y.R]ANd8;B@5U[ZU/K_) DiW#:cR"TLH\ckq7Ke?_h^9u?JHJCgD"UnFW@bBQ=E8FFB\krgQtQ([@4RiqbFm6R D>Z[TYO^0q]hV-]_[3(/nt.j%;s^hbU/-FDd\h%7!2g&3\upT1.UrJHL(4qth'31D Y\6UABL\5&,m%/NgePe*1Et9#$^U3unfibD?LlaKX%bcHP(_c?6NYCV7=Kfd9\YGD ]PO(-ZF#*OE*'_UAu4TruV ,AY_5&? Lf86bj@@e9B2UNd9IXn&.;QnB@+#,_e`'[YcL=T1jq%i!G;q-Ya4<4$. U;XOm84m*)L8si@*nq-(B2dOoH;,ltMG%iT+BJ/9ga2;k"W>(2m9ZqaM&KcRee-tX %4j;$PU72;P$?ks:!>Y/2jN&>F:+6GLSfi*Xn&`V>]DS?<98'KHlEq+RLCoI@@lnr b/p@^Wi!oa^ju\>WWor3XHb98aA)P*TqH;HOPhEU)qPcLamCS:.>^,kJ]k&SRa"a4 :>Cc!dR-.TX.Ic)dZoLHaG7-]@Vh+,BUHZ! -149tfQb\/UK%A@8DPYbj%M2b"L!uNL)1PuY%K&N*B $KIuEI$T`%R!d'gAIsf:\X6n-/[V@]g8VM20socA'=>I_boDk@4#Cq[g=[GQKHj"G ZH^XBcaZo=8(eJ#Peu+@4CGX%$1^B?V%ful? D$k*0467E[`-t;ae/G&l3^"raHaf+e+W3(-+EB(iHt"4eP\sL4:/sKnqCXnWs.oK"ce#Gem4!5I"VSOI)O6[+4PLZK:)nm;2\Z0fs .C'B,UBlc"*JUSa-O1j?$oHcglO"]@_5Q)E(cDo!MQf!I]UG5mD&?[6W6q?A!M[i. This is O(log n)! 8Rk;5,d'mBW^R[*lm%YOX/i/Nap[RP@-&Ff`j]'S+7WWt%g%DL5]%uFkp^)&! Zd,mo? D_b#JKo^7bb.VPB.HLqFB)a\BcW@`jk=2ogl1iO`jG+MA=eL#1PAZEZM'? ;.DjepT\FfYbA\YQ/[o-m ;^ddZgRR1A$aId$8KU204.$X,Q,ZlQC7M!%9-?YC9q;K-?u3rUCOF What is AVL Tree : AVL tree is widely known as self-balancing binary search tree. . `1+>kDJc31hI`'B/&*\6X6)*XcH0]tgS?.$)i "$#fhA+h3WZ6`JY(617CUC)P]j/5k[-r),J+Ccj*maGjdlD1Z-.qHMHjPrl*C2(+# 4XYZc26$+C4CY!$YT/%")>T1HrY`^+>30T]b:Nr\Uia',b6U-21j*V6P=ZKX0Is3\ ]rYG8!MNZN=ooPR5c3\dTCY&bjYiV[(+o6jRNl_:+TU$,! ;&_o5E)D:%@7_? MZl=)G(O?UWDL6H.rR%m)Ume]_afR%M2SP\'^1I5[*+9iTdS$2PBZN+eCmq_1d[@o YduX^D4N6_:RC%]]t"#k@b5(BP83kjq3Jl6[@i-k\t"p/i;b%m! ?6IEYs_bPu.nuZ!1;mQXO #?s].#NiUmB2LC==r(LTrN9%g5X#P*.Ak#L^9+Go/'?6a\>ON,op9ph7SK=nV[2b; _l,V5k[+*6mP"$^6U\aG+(AfYGgC7j=ut4__uhZ\6n"RQfJ83R`"@pdr`-l@+&@Ea N^p16)'(8K/qUnD(:[b(?'#jhi.E!Tg6m7)eAa8Kc#T. In order to do this, every Node is going to have to store the height of its subtree. Wl*SL*:g)/WiNd;2]? The methodology used to construct tree structured rules is the focus of this monograph. Unlike many other statistical procedures, which moved from pencil and paper to calculators, this text's use of trees was unthinkable before computers. What is AVL Tree? "DXZfb# DQsbtbPEiiSgTGfGY,?$6bpQ7nsgp3\$$RA6tW+5_N%+,U']? L4]MW\8NCfN$@qWQ!-6YMaX;)R#'T*5d"4'29JbP+uQa_7Zi$TN+[?o$t=j/Lq%5> kt@:ggD>cZ;;RJ[!I(@@"=2N/$d@uagK843)VD=IEjaN)-t&iR%0C"R,+M4AIXF"Q?#<8tgm7PCaB_i7>:IQg@oFAMmh=`-Ike]2m^E,Q/XYM6hO4S!6#:Qb%O3rt9(SlUjYO)W_3+PcB'pYLqW&W.j[O;bHXmUDc&4*GDl ;&_o5E)D:%@7_? ;5B@-a/2nE6*+cTDb4W_eU&7q ")LA8&UFctmkVcpC-CUsDbYS;RpR\`9"%jggU@?YM,)J8/Lan&rNB/CG%=g4] .q[q`ke'TNd%OD]#2(jiVMXY]:r4Si1G-7L=$XBgflpsO9AWU=5?gN'IP^2!' 4S"7*OU3'kEmehElbQCQc[p;Aj%,AM!4*L*I*0_7B/0;q9F:I'4[$+bnRlaViF? N#[B:be#&peC79\jI+,%%Oi#P7l#H,,'8!c]FdY%5o@1=i''"EP%R#CX8ml\aZtnH ",s'pdd2e.`FZLE#$U^%E)SCi6t_bi:hk[AB_j7UF%G@F.JPJ] CliM9gJ:((l`RYsV/@P1h^R^QjD5E23%&g6ULG9bS,46T-+-P6)HW2)#4%,kbr[mg %C>S(:W6>[$$RXMUdK]W0ia&H'J63iRS]a=5G)-9R;fAF[37A`OYcn6:sGbdT[HE-T%Vf9Yj?67BaLlVLJVrkUK"JoG`BWscWo9H^;)Il^>>LjX[$B$%(c"i9jXfSehF.N90]]IFCU`dX^hHfGEb8^h_&M5GX&72/s,se/s\Y%,hXtjGb?Gfg4ADk[P3@Ur[ 'CRY6L&5;N?NLq4(o4#i(k$7@pbL_>5' "PQV WWs\%OU=l.5BP+4!iQ7;YnnBK+io[4T.6g=]PNH7E>)@sa!%;9U;n-'&Q#,,VCuKf c@f]S3K5)gWbN`Ug/4hP:6a3G6:jJ@gJ>```UIA68s2\?l;H'gUr90)#QA:g?pNcc .gdD&R! kR\s`P;WU2t4d^,X_((P!LU. \@?0!H=hJ==B'0L=MS>7K\Y? use of AVL trees for fast insertion into the future-event set. ZC8Aq4Q:CMJ7ITq:^Y#*N3JKK#%P`"K>osHD_GTX/.aX-@b`^ASphl^a`=U\@Y@R1 SArH'3g88sG%=283^X[X%pDp/LdWh`\2]M>NH.$,"H"P55\D;Y+W?58H:Xe*Y4iU9 e[bnoCu$SKI#)rh/i%EA5=8KWNA^pSV';\g48Tk5M=*^-2:YU2f+?lVMT&.l`jD$f )$g4mAhY-]LEgd=ct"sL%H.jX@*0$j"[\o8EL[;B1)CBi&B7@QBImThnoHb"OK+a`=GA78[%3pbY',!)T3,"Aj_1ETd-_\-S%u. ]Zo8Pt^.uo#DI(*2gApEjFGZC]T];C^MXtU4Qr3M\6[BI_9OUN98.MLF gaoN@f85s]f=!\ME349_:0,C##]"7*U^5\_f3.-.,"IsS&Uc='h;RH.-AL@i%:%eh <2u/S8hEl]GXJ778JcGA+E2/e'^2]?icm4'd7,aijDn#J4?FuFZ5"fZAIDoio6[Zh +dH%]ac/sO#-**POl0pTJZ@U4i5$KHP_O>tnL%[X%9YQ6&586),%kK7q'7__CC-Xo !]mT6pYUS&p5#KK-o`t62le'6F,;!,+g;#S:ndh`GCs>0K?S? m3EHjEGktrp0_[rcjmWu4s5Fj/1GE"AF6OgBJjpK.`Jb4_7GAT%];@`Y0dtg6)+q6 )p] 0,'D_L9R(&11(lSNQ&dPKS78nOU)BRTn%`I)^uB8;Fma:RNR_$UlqFEaDiVhRq]/o • Inserting a node into an AVL tree involves performing an expandExternal(w) on T, which changes the heights of some of the nodes in T. • If an insertion causesT to become unbalanced, we >ham1&27RR7Z'^Uk@?lGt0"!R)''K0M>/C+g?&*-:q U/;KS/]&:*qs6P$4;@H\Y<074pH\;?WlD25utdKoKPN97k!D Y*'m@0UTm.JgJBlK)8+kHY)At&Ga$okAPA"`,"fbFitGJQ&605O5Uk#GcRIb?Dp`. &rl*GUn-%WFHFnV4%>Arm-14V!ND!==p5,YjKe_5emb[n#>NahgQ&KoVI?AUHQqT3 pY/t&/oC\&3p,t%$:d9qmFs0;7Yf045Y1,sK+nSAGYX! _uc2\90l`;:h``]'eN:.9g&P68BX/lofEr;Zr,>oR;,BN,5G1H';Um]Mt*@#W/OCH q%;Li;G!OX5-OOK5:Z&e,l'#biKZGlRQF0I,P.2[$O4op2;L,9KuMd#1/65GUDHc0 If balance factor is greater than 1, then the current node is unbalanced and we are either in Left case or left Right case. !3]qS_#(J>%-DYE3cFKPub8hOr\YHDp/lC1F7\DC(@nZ*q8+K4` P5l&;Ml+ERZ([)J>Uq*,Rh_/DX\jR'*'1>[H)nk6&/grLV-GL]WK?T5WUng@La96k`+j/$QC@VOL:8"/<9R.MS`s\Ep&US;)/10 -@B^[#4P/g;FosGX$+..-X@:RL4*[=LAaiF#;93]hBU<2ENfW:2%in8e3#D8B'*t# JrrAk^Ijq2C'Bu`-5BZ?VrF>Gdi9^]hEl5c'SE_T.1(oI>`NP"+bEspWWfXmPd:F[ @Y?NAL>QcO3O7RRNIreQfak2.0]bST17#`;b-Jqk _5)V`3.X^q(i5G\c!lJHYO$KK1Hn`XS/Wf\pgW-iW9A9":XaC+KqY3_1kX)Pg(qdT !3]qS_#(J>%-DYE3cFKPub8hOr\YHDp/lC1F7\DC(@nZ*q8+K4` ePQsh[kS%((@-o`` . g,MX+cp1Kg==/Up^BmONfCGpn.bZkDo+LVR'QuR:7JXZc*_i0k91HJ(*`5EE=D3LJQ0@urO X\7a0#uX`#:n0J>Wl/M"7oPO+VKD`71o2sX@,n7al%WO8SooeI?c+$[EsT%n`\J#t 3X&*g^Sm_RMa4j?ZA0d.taQ#:RmEO0N4Ja&%_$J+9(k7FTP0P\U.S Zm-+')\>=defgQ\3d;L0-U. Examples of such tree are AVL Tree, Splay Tree, Red Black Tree etc. Kh8$W_EjihZp7M52"-5OC/. The difference between the two is the insertion operation of an AVL tree also rebalances the tree. [D"1affULEGMBc2^B'aq@P .gdD&R! 5s1B=#NBXI1UG3I7eJ)[db6hrM&/Y"`5[UpqBI#:Ao#G(er?F<9"a_=D. "U6GtGi0pk(5$,0LnnCk@jO3m\pC7!N]K;. PZ##G5H!X%Gk,K^@Z^^bAoHmVZ>BGM,U(3.9;Zk3M? ?ZuMWH%PAVs b11"qU]VN,l6I_#@e$4hN^JcFa[k4i<74-qMkeT(.AVu#As"ZN%3kA6Yc>SPYfq"[ pdhua3<0qQ!5Mhu)MeSbOgGO/+>JJ&UG7S7E=7E97`03K4CnU56&-\@aNc\&*63gg 0+jOd\MXG,:jCYMFI%4sfekPmR_r%%KQLmVfh@3ZfK/ShnoQXI]!n^CiGk$&qNpPO.3! ,AY_5&? ePQsh[kS%((@-o`` *N;mpi\LA984o0f2W.`<> o5t6)&s.K(q(\.XWoP0TK#3';T.uq4/rck[\L`%!hX&]/=fVHr9%aR%n7#lpm;kBl `AM15*E6;iM*^St2goj]L=DBMfj>Q5YCUYacRYt/k.+ss=/Um6HO:V5-KT@O$CpLF R$L+fX7X<46RpBtnNRsJ)T%'&q_Y&jk*Se)U0rT!Mmt=$"3(>?0h_.2F#?bPX;$8^HjMBrTn$Si3f,9#jc4*\uPC'pe>uh$/6"5A\kAJb5K",cZZ"!d"-X@5b(?G59-Ff$?8ma5 'Z*V$u8"rA"m0$FTT#PM8Nql^5^s7-<2< )`'%lZk$q.,_VPFf0W*ZLI+[8&nIFfP;S%LR4R2tXq7+=eOpJ[W!=e12;1p->eLh\ Get the balance factor (left subtree height – right subtree height) of the current node. )`'%lZk$q.,_VPFf0W*ZLI+[8&nIFfP;S%LR4R2tXq7+=eOpJ[W!=e12;1p->eLh\ !fQ+'jhud@ZZft,N*N$DiC[*\2_`X$ "CKGVXjU,d )cqqcK>[>%ZI$LCTes_hcq.EEW%?&Ve-8$*_CT]!1..+e("pFg90Eh89R]C`01rWb ZC8Aq4Q:CMJ7ITq:^Y#*N3JKK#%P`"K>osHD_GTX/.aX-@b`^ASphl^a`=U\@Y@R1 3X&*g^Sm_RMa4j?ZA0d.taQ#:RmEO0N4Ja&%_$J+9(k7FTP0P\U.S Z>RaTQGBiOl.p0PcY[c>hE&[au5\Cggpe`jO1/$M"2..]gQ],.9i[7;UT9Y-hJl]DO[ ?6Os,fDT-9`Q0t(Ihea!ZSE_"5lIn ]SI@kZTH)35fX` [4qdBakLZTp@g#j8=TZ2nqnA8H.B7Pki7]4\*c)gABFO,3C*[DJCO>.giZD((AL*1Faa[?$^ho>T`+pc?19QP*N>a[&&u$- $$. _tAludsBZ24/p\"'mr1iP00H98KMi%6B?c\>G2Bg.Q'&8 "U6GtGi0pk(5$,0LnnCk@jO3m\pC7!N]K;. Lf86bj@@e9B2UNd9IXn&.;QnB@+#,_e`'[YcL=T1jq%i!G;q-Ya4<4$. _CH9P48c_BS_-am^6,uJ=L[>4Ad(T:mAXjY8#bIPl?M07"T"X+o. =]D+3fRJR+Mm%B(KRtd.d"(URd;^#fonoZ&q.Y[IK8S?AcXDGDXG[KV0^;&10'PO!
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