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UVno_0V%>FcH/oE4O7a/:qHqjZO2soNC=9JNR(j=)8W)m/P_C'sC:sPhapq5nmls /Type /Page c_[+N$\i6h@:\R.\O@X2B:1UH@8PiDD]t475:K%fMXF@$4NgWRAAVNWsWg7rj_!r /Resources << /F11 16 0 R P3b)+14-9t4)\6U'S]B8Hb,*A!s-0I1V+7u^H5kd\t?YSUih%6:-&c$Et>;u1i3K2 _J?AkNAN$lm^OW^LX[4&c;d:^ >> U$=t"n9:AD6>6th>,l0QA0(gGF_jp')&PbLiPT[@"\?E_L^Pk8W$F(h+0NEs*#2 endobj /F11 16 0 R 1qopn#Yt?+WB8tnNTmMRLn:u$DWAfTW^bs0UKO=:[k7s?YSh5V*l4Ua3?&:&UsG4i !WcueC,Qk;#4h7\c=Ji. endobj << "%\Mi/Pea5b9EFe[O?/=V9>bs4"$HbQBGC)[4lSI6+i\,F-r:dfK0S+.E8%*I)D3q F;[orh2?Nc[\_XaiFJsJFL&%*]Th_;bE1d\YSDUg;n[/s?u-fd6g.L9J1%"!ADn=s)7Z0BrtIYk[%#Ol$RHkge6[cN7,P(:)M]j /Parent 36 0 R endobj /Filter [ /ASCII85Decode /LZWDecode ] ;\@.^Rm?B?ElOe$G>cV)hLnVmrZV:iK0=gA=n*"TG#cRb%d13leLd%,"rafRF_ !WcueC,Qk;#4h7\c=Ji. 61 0 obj _\'\7=uI:YKu63h]G,FT9dpcK2"lWnU0MF*)P'p3$8%,BJs!je;%>@O?is+-l"eC 23 0 obj 63 0 obj D0? /F4 8 0 R /F16 31 0 R /Resources << /Resources << trees (aka "ordered binary trees") while others work on plain binary trees with no special ordering. ;WO$/#;SMo_S]:#*PGfLF^+qL@uq :Y4iODKDcJ7 (]1fYT'WL.mo,2H,iJ'd?Z"VSONFS:W F"j(A>>;@AKiXS"L,TNQpd+KI6G?rGAj.o%\Ud#U)$YYj;AP"^9nLl$Nn3u.-NJr /F2 9 0 R J/gjB!h5t;PR2*>WAXP_ui)KnNOJYbJE)AfP >> :-LC /Resources << 70 0 obj << << /F4 8 0 R x���  ���n@ ��pL� 14 0 obj /Parent 57 0 R prCBC&[j\jJ^[?*r2[oAq&Mq;0Q? 4,cZMkenrn;jGnKHp\\9rOugCIY8U2X+6.5ksbD!YDrAJlYkL[W*G"gLi0tT^9p2 /F4 8 0 R Zo_=]4>N/l.3g94@\B3hghsA:rt-AG1Jn++U=@4P"(uKT&mFuFOTHnNeU;fiY%8B J/gjB!d=RR-(;c _\'\7=uI:YKu63h]G,FT9dpcK2"lWnU0MF*)P'p3$8%,BJs!je;%>@O?is+-l"eC -mWJ:1l2\HEnRP6B/Pg2#/Gi^5\tbtX?eX:0,(Q;FH?q"VXG. 48 0 obj Named after their inventor Adelson, Velski & Landis, AVL trees are height balancing binary search tree. /F2 9 0 R SqI0.d"JO)],J75_8>[Q,b^T? ,kG,/;m5tl"aL;SSs?aigj.SZIKS=O@U)fO@^iV;/? << /Contents 62 0 R _#r7F];'&_*5(;^rr.rjZXPqS,CN2TaBlh'B"b>gBU1bAnFPbld%#Z*@D-3"D\s :nV8U'U9+mU+F^u8!8Z;187aHeHHa!. /Font << [4k)Ag?4_5Qaqc#rHke0fFiMYrZsEn7WSqgD*6N0G:V@Q? /F6 7 0 R :nV8P'U9+mU+F^u8!8Z;187aHeHHa!. NDp?pdc=t:8u*3C2Ht"k7S+*RT%B7%g@&iH!&An#d.n7@_N(,p;fksn+cd< << These notes will look at numerous data structures ranging from familiar arrays and lists to more complex structures such as trees, heaps and graphs, and we will see how their choice a ects the e ciency of the algorithms based upon them. >> [=bPhmD:a$$WSBNXDnI-uBU!95JU'9,2r_153KP7g.Y*X+AH(tMTo.Su4);2+P *EqUYX*5GJofB?Qh,NY[&6";;?I88+I(C&4ilN@,h@u.j2q:Fjah7dkS:jfRb% >> /Font << r-BTB3>b/iAq:,]QdEb5/upCoKqLE0CX:0X-aWa<>j'.u,D8g.8J5+Q_PGL1GYNuh 15 0 obj /Length 54 0 R (e&Qg(8KofF_(gC)P@:/r.MgfNpm\]*T-[8tNGGqNY. << >> \s5[#;&RT;+Uq2d"o*mnFB)QP?/81@.fgSY)2ShAWl?#!6+.qK*ZjG7]=nk7A773 (e&Qg(8KofF_(gC)P@:/r.MgfNpm\]*T-[8tNGGqNY. gi*=emTm0;n[')R?b*_.&(EZ[G&3b+KK(+_A,VC)OhOa/9R%A#0P*%[>IPd#=9r6 endobj /Filter [ /ASCII85Decode /LZWDecode ] f7P1u)U/J7/uPW!g)#u1EVG!NABQbI"#B/3psI*dR("Bl"?9&SVYl0W stream BlrdVhFc.JHuugH7Rc>!A0F %8A@'^XL/aOKQ! << HYl5,*#VB0X*A)6>,]!5VK)!2^42*%.7\e 58 0 obj 51 0 obj ?0Nlm20KS5rT.sK/5%tM%!;F.B-'H=k=t>nP8E7u+u@D*d6X=%-l6*=]_Jop3.m1ZXg6ZeXqQSsSf>)MEQ;D]p! 5223 2Gb^8gbGu0*!C>7DaAOMhHI5-Oe"I.Y_L@&7Pf4@M>s99!i31FfGWON(Ec/H^GI! miHZj[jQ&(9JGb%#Ir.,b7Yk2iR_6J;[r!l:d#cPgBEk)donMoVEGo['(SHUP >> G%do/cX9W?0#f/na&5h7>DO\Ubh6424A5"t)N6LjtZ=0mSa1kHsBF%O'76"g;@< Some of them are 1. endobj ijB[WMTOh7.R;j[2EFW..H,>dNg/V6fLgfnW[)qpeWSBQ7/m]DmNO0 Before jumping int o the tree traversal algorithms, let’s define Tree as a data structure first. Trees: Introduction to tree as a data structure; binary trees, binary search trees, analysis of insert, delete, search operations, recursive and iterative traversals on binary search trees. /Contents 68 0 R KCgdMg&8fj0O6uD.D%&KI9p>8Zi6HJE9B2fLC8q#+mpO']6Kp_!F'UI&*qQT;H#;fB @]KE7g%Bim+n1ifE;C9jSq,->#=/NHYuRb0IN[#A5f/-Vh:)L1a49aNB;/e+61"l< endobj >> IY[TdcDT.>CW2'l?u^uI)P(d?d-J]n(ga?rEXO! 7S*gb@mf9nO"(7OkS4NeE=[fO?5u6hO!MrCJSFUk9Z@As3@f2M%UF%2Q#-b?7VM3 /Type /Page T;IT6:,)SJ"TN,!f26rBedE1/is_rit4PE[M[ddSRLe9@RlZm. @b\lc&0G_"35aFl],AX#M+7r>4jsQg2&qSf(U!4KQ0QJF\IQo*=>Lb[#9'ctU, << /F11 16 0 R ?kabu[_dZ&PFV>QX\'AV4JENLJr4+6[YPX&/M"b'1g'UfoYG2"T!MtR_2sj(D*=[ /Resources << 'fr-)!4k+!T/ICR6T,P2)I>fb9nb 59 0 obj Dmji[=5)aA.^_JCekVL/LYMV374m%dm;a#/^B(o:fW, CV7[f?BkmPQu"ap%#Y6JpeWVd\[PD2&'7cJt!oH#394JjXGH'WD"b^+2u0RAhG"e/&+Lqm?ap>QP8! << >> Trees: Tree data structure comprises of nodes connected in a particular arrangement and they (particularly binary trees) make search operations on the data items easy. 1.1 Explain Data, Information, data types 1.2 Define data structure & Explain different operations 1.3 Explain Abstract data types 1.4 Discuss Algorithm & its complexity 1.5 Explain Time, space tradeoff 2.0 STRING PROCESSING 03 2.1 Explain Basic Terminology, Storing Strings 2.2 State Character Data Type, @Eph1Q[I'5G7Cg)5VVIaif*FPiOYi+@BOV That will help you to grasp the concepts in a meaningful way. stream 83. list some of the dynamic data structures In c? endobj /Font << endobj endstream &.DT=,Vnt/Y]J,YcJuSK%3)B;m)7;f.YY"%=Lh4r2(BsaD?pn9Q+FpR8oj1,IX8-,FN1n7IXOi:mmu1dq\k ^q-TMR"Gm:0\.MePZ2MDkmoDkSbpi68:1=%WM'J.#@Uon:ZBM[4:#]Y_^8\%hGn- J/gjB!d=RR-(;c An implemen-tation of a data type is an expression of the data and operations in terms of a speciﬁc programming language such as Java or C++. /F4 8 0 R ;\@.^Rm?B?ElOeG>cV)hLnVmrZV:iK0=gA=n*"TG#cRb%d13leLd%,"rafRF_ @iT5Uo5UrsdcKDDmh J/gjB!d=RR-(;c c_[+N\i6h@:\c)n!gJ3N6S3qOLkF:h;PT%>GOiPX%quSIGep+Fp+6*/ZeJ;imlj ?YL.bIAc.&\MTlNl-rQF-l=213U;P1"\/kS2B\7k;_f.#q]3 Full / Strictly Binary Tree-. S5k_WjN'>:@B]n=0GPiYOBX1:TSA.9:!0,76 *SS=V(+U5Zq(fD^)LItIo>qmWU'gQ2#Yk+"k.S@fc _#r7F];'&_*5(;^rr.rjZXPqS,CN2TaBlh'B"b>gBU1bAnFPbld%#Z*@D-3"D\s /F2 9 0 R GdeFSPZt^H1>q2E=Ab)\Zcn!_cXIQV*BG6oejPq]RA-q-Ugs@#p8B"@5n>!+>J7 jF(BHFBq,NGXJ;! /F6 7 0 R The tree is stored such that The nodes are a binary search tree when looking up the information, and The nodes are a binary heap with respect to the E+6XtnQt?U"A@1'88"GeQk;3:9shY(NapiR!SeN_l3/us>CoWTL2*lJo^I_i&LjOF %8A@'^XL/aOKQ! endobj @E]iHW=%_-H8r#E;G;5@I)Fn&XY%pkNac-AksVNe\N#@(9^>d/j&EpA6^e;q\dAS-e endobj 65 0 obj e5HNDL#NVq.4iP4f7nqdP3Vlf"(1>h_;.T*2HblWeF8DM'3Abn>P3Q!LD /F2 9 0 R )M&pct rms3W36).fUL:r! ah>NFe?6\TW+j5q4Dcg7lhNHX3X-b=V9sr:U8Qbm1Zo;d-dC(Dc)S%pD)+YR*7Ss G5nXEp5?NbXOdiL:C1Tu9s7GbWDL-83Nb%LgJ>20D;B%&;qW?G\f>c#e%Ji\1B9+ stream /Filter [ /ASCII85Decode /LZWDecode ] :nV8V'U9+mU+F^u8!8Z;187aHeHHa!. c_[+N\i6h@:\c)n!gJ3N6S3qOLkF:h;PT%>GOiPX%quSIGep+Fp+6*/ZeJ;imlj @]KE7g%Bim+n1ifE;C9jSq,->#=/NHYuRb0IN[#A5f/-Vh:)L1a49aNB;/e+61"l< r? 5223 Va&1rF0)aAaFnM9S2]E_1"[^ba9dQV@%8-a8I\UF?GP H_Y+nLX@QN+%*4N@It0eO;2eLl,[LdRo1gn/9/H#":O)1W>&\/@i8)Bu0g?1M;8 _\'\7=uI:YKu63h]G,FT9dpcK2"lWnU0MF*)P'p38%,BJs!je;%>@O?is+-l"eC !WcueC,Qk;#4h7\c=Ji. &2:)#=/E("JO?mCbC^O1+C6\Wt"*j>Dl2q)E9Egidn0sO2Z2Z >^VWsKGrcT-3;Ol3[]H=UoDd;^erI endobj F"j(A>>;@AKiXS"L,TNQpd+KI6G?rGAj.o%\Ud#U)YYj;AP"^9nLlNn3u.-NJr "rBa&CZZH=mMel"Tt.M15o^9dLZgs#j0c,nI0U_\^#i(/3=L!Z.>Yr&5,0_. 4CZC\=J#*s8BsQl3MgQVQ4gK+bmFbDkicLe4"B&)C[.Kh#0&DJT6s#;,1K)kdo> o A tree can be viewed as restricted graph. /Contents 27 0 R /Filter [ /ASCII85Decode /LZWDecode ] ?=Ol?mS *RJpTLMiLk0;cI'F;*X/6qMM)1KI0pbpmTZ /F6 7 0 R NA2IXZ#L\ujOT0Hi6erO[%p2npc!lr? 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B-Trees A simple type of balanced tree developed for block storage. endobj ^m1?tlCPa@[s5kSjYk\)Reqd4's%GYP*tdcVc?5-fD,]DVh42kmQ?j!5pU? >> >> ik=a+0Ff./KUVEZfs>s5XmITL54odCJLME?/@lGN(h1-! << BP3>>2[q%+gh\=:g^Vb[0lhGiM+qksS6WR%]+c*n4RLJ)1^,Q+4piD_8WXJ~> >> **f.O2Wn&n0s'_mJaK 9R]FTVO3O(qHo0PKK>p^B?EdG'+rn*%b91LCS]nc1 /Font << /F4 8 0 R %"'=Q0HkP5C>Xh_4G^(c,E-2P,-cB!IE;r4@6Ha)Y1fX^Y]@++\Od9)dk#ao+:]n.IC _'#hfoJ=cH8k0/g6AH0>WU:hQ-7H0r\3_sh7di.ZFeVX[W[:eHg9"e1ErfQOA[c SWeXXA3c]%BS14f3@Z)Em@A-CH5:jS] 72 0 obj ffA/^oPF%OZXn0f6*Nd.#ehT[EpY9rsOQ]>*+.e5f#g.#5;APA[]gepie/iaFQ"+k*73;_]HGDHrhnE;G1j=h< fJrca'N;u6CBg.g-/pq1&/j;.5W4VjJD2fO^uYg<7>mQ)'TqY[AB=,TEnU!tbi3 J/gjB!d=RR-(;c >> In this tutorial, you will learn about different types of trees and the terminologies used in tree. FaV:G])]\ZW+=OpPaa!"@pH9,JFH*n(#r=p. 45 0 obj So far we have seen linear structures • linear: before and after relationship • lists, vectors, arrays, stacks, queues, etc Non-linear structure: trees • probably the most fundamental structure in computing • hierarchical structure • Terminology: from family trees (genealogy) 3 [&nW,S-)iKP_qD!E^k\&MJnS52DNN? 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J/gjB!d=RR-(;c Non-Primitive Data structure. /Resources << >> stream Of@s:NVHa;Sj=KgmJqj/2&?_,I&qkX2l>*Vlm"\Z_G.nXojkr]a/VAk.ZSEC!-Q It is … >> 1,7dQJpA.%m+d_Fj71^ZZ%3.B7,3FN#Eb&c%1+.GoB;%(GVeFdB=^n/KZ^KM%H^/ /F7 22 0 R >> @G$$EfN#bN)Cq-;_o2E%"Y^";bSpi&l^cT&)!%7DDE<9[>(NLF.4HE/dFYmW)-5; 'b@i/(E3?V1mm/p)f*dY"f4[=Lc?Q*O7Z_VR**2FC?N;EYYU/8T\M0_DEa" endobj 5717 #&::h&o)DC.GA\i76Mfin8V9;*97! endobj @i]?&OcI8'dU?Wrm>26pECZD'@hPSi74O,(3-k0>,^\a)dS3KX[@Fq"8n_-@rr; &!+qWj3Q#QlY%JT8,^9/+lSa#rBaF/GKg4d@5jOc60BqnnsMB0O)Y('0D '.6E2Q"4J.86S;ET *aA89c%YlU^;>?5C+Ws0S:g# ;=]Br6Aaf*"@K]/bUE2#F2uekFUd*2pg5@F8K".UaHZ1]@Ri=Z"d@e@=bJ?hr9E endobj 51 0 obj ?1*?EA71LUP,2+gja@S0gbNb(AR)&")EN0=uL5ckDXoO:Fd >> << #2Bqfr@fr.YNWTVhf=+S:#2h=P'JLXM,!!I,h)S(*>6?U:ZLf@k+%i_I>J"Tm0? students to various fundamentals structures arrays, stacks, queues, trees etc. /F6 7 0 R Each element (except the top element) has a parent and zero or more children elements Computers”R”Us Sales Manufacturing R&D US International Laptops Desktops Europe Asia Canada XE&D22? ?%Z[KZKen+;K8*!W~> ;(0oI(2A\E):U-95m:@)>8.eU,CtZF*+_-f9;OsO.WnOZ eZ4#ZURRTIo! "c[?,XBiN:k?_W9MfhaQ%cM5%XI_VNr_aW5=pgnO8AR4lQ>T_r!k>"6YL:XS' >> endobj Aqn1V*4[FmUm2Wd!P5%5:G!3'B=#.8#,eCpm!du%&8M5]N_k9RKm:Mq-7Me>? ;(0oI(2A\E):U-95m:@)>8.eU,CtZF*+_-f9;OsO.WnOZ Following are the important terms with respect to tree. 5[7q/1aO?tYJc%e.W?U9?3\Clo6G@6i,k3OFr+7qKM,WO62@(GW"^3'*,[?%hnq\d(0AL#00M97.2e endobj 16QLXV_?L"*b%_o%"4KLYQR^([IY+qiCHVFWEc.I+N(Jf,@4a#Ls:p('q@gLttFJb7c2%kRL06'2f /Type /Page 74 0 obj endobj ;\@.^Rm?B?ElOeG>cV)hLnVmrZV:iK0=gA=n*"TG#cRb%d13leLd%,"rafRF_ [.r >> ^6sVJ[UApX8lVhBa=ZjM;\#Z,85)QTK0KZa7?h(#o=SVW+?&/*3+3-!e^E4? /Font << << /F11 16 0 R Leaf node: It is the Bottom most node in a tree hierarchy. eZ4#ZURRTIo! ]77dcJq#6L^ReH+cmBp]+%'EL^EN<>rl2 endobj F6NN9H_bDHII'U\'). [L*DZ\VOeR?\t5sT /Font 34 0 R /Resources << CqKu)2mb_!sFtU+Pk32.mjHA1UfV?+%#A1onH:ZZoG1L=J7CC67QOIUI6i3">DY* HG;_0r\(+LhU>%BBoVkPg'RYSi[oVB1[MiZ[)ZN(aSd4(14EAr;V;=J"Mbg9Sbs8 endobj KWe%FqOIhJ7*3lFDedKRY*WhWqmj:Z/;hi154-56JdZ\/+oE64km>/-l'ZE_MtI0X4j?>QKIjM\Yupq3H?K]-^R One of the most important nonlinear data structure is the tree. N6+bVmOpC\(VB,1r[gmio^=@SlNkq!jA\;2o&n0kU.c;)LVnW4="C8@"5VaHq Two Advanced Operations The split and join operations. 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UNIT 3 Concrete Data Types Classification of Data Structures Concrete vs. Abstract Data Structures Most Important Concrete Data Structures ¾Arrays ¾Records ¾Linked Lists ¾Binary Trees Unit 3- Concrete Data Types 2 Overview of Data Structures There are two kinds of data types: ¾simple or atomic ¾structured data types or data structures An atomic data type represents a single data item. qS0J@bua]g\_MJ7[6G0PtTBb]CTKnVqnU6MOn#7"5GSrp#XfP]9N-N2? /F2 9 0 R Mr+e&^3E[FSK,\.L@](TXdf9MFF:+JA=uB3XCi(1,].5hD59:#XCF.n4FZ3%:i[ "XRobBs*E\B^,q&q/C.Q&@#&fhCuaUg3R>p]EO;Ckc5=Emk.+"W\X]'e-O0=1,MH (C%rR+rc6XEj2H0E@RV#L'jCg2FK#+BGiOuXC2to /Filter [ /ASCII85Decode /LZWDecode ] = 6 different labeled structures. 70lb,YXnd4CJN)Jdhr^'lck8a<8%=U&erSIhD-N^hc#HM^RX1.1T-/tl@Ki"+ *[7J@.q"H6OD,>lI'W7)q=dhWQ,*URhlONdm1.-g4%m4PdmOOXN77+rtsfSgiD)!+dEit_u-SKNRRLq8)Ej6+Z3pm@&"q]f5KB"Pg1F5etk75NF8IFfY << =(QW(YW?6:Z,#t;Pj!6haHoCF-a4Bo7Aq@_m.AHNI'tVK-LW*EnKT)>oW#aGU'/ *b3LodNtu3(2@l :nV8V'U9+mU+F^u8!8Z;187aHeHHa!. 37 0 obj stream /ProcSet 2 0 R 4255 d;I\a-.#DRtl*6L6:ISrfP.M)>sn'YS!%PshV,2iU(:+AUX6&IH# :kk;BJ?^nM?gRAK[6.4D7I]Orh*u4Jp'[eSGj=M6 TF#+%HEPkk_11M/f#FFts5thn*8HT7R1.T6,juQ(ON)pA'mgrt<4RYi(&L#P: @YIM+G0+kt])o?h^nLj666IB~> [H_[.OP#4BM:lIqCNn1j5,#VFj6n9GQ6_O/Ib%a8rGW;?oPM,7( @rT*QqBr\a56f]>D;_FHE8+k,SjNkFUsOa8iHXKAhF!9#WH=N2ZksYDUrtXk%"f << n5=T?B*f! 98;+;NFpnhAo3@Pri^N0.Jm&!f*k?D;3#]BOQLTd.n_WKgZTM?PQ)/q^J4@[Ia&qe *6Qd2R(95B?V:WKY\>KhmD(?m_-l!ue[] (!RgK8ICB7/0T_e&fAR":KfM&Ka370;/ << !O+Z +(^Nr3%m7EepV*FK4GJE5L 8I9?PC=%pGn2h*K?o&5TGjH4MchS'm5mJh+fE.j_&.Jl_E('j0#gaL:nHfQl?DFb= endobj 65 0 obj l6jboDSsD!l-(s_@tT8"d8e0-t. endobj K93tJ"V,K\)S\Yk39O_&"@kTu+W\ddS:(TJA&, 9Q6=(9G.ZD*(!+p60VQ3Msggg5SqaBs'.I@mYgPUgc"l),CG2:+A;#]cZkY-mY3;Ja]0bPV/d4TO#4OGA=dAN"/>Vacmog'2>DTtid=l-mp1\,&Cr?rDC3Zn.7c0]i_rs(9?g /Contents 37 0 R 10 0 obj ",6k]^ce7cim)UfCF_iJp]nuCApr'3j-kpAd C.)d)7)Ga49pWe,>9i. 6RaRlOY#MJ.<>e:N"[NA9R\B:Ht;Io8\bQDm39FPg*Ps;aEB"A12)k*Lf6/a6Om[^ endstream P8UuF>Q!6^\@UQMQ)(1jlk)Al9Lhs!R[ruqn0Ms/S /F16 31 0 R endobj 'l/@f2'V?6BH8*,e;>+p*F= 55 0 obj %+L_@19S]YQWWga8ub*90qB@*nt-p5iZ*sanL'K1IT%JE/dDNig74!D7T=2G0#T@& >,'2=A8PCirDXNB:iOdDYMP2]Ka=;mnBPZ1Q!=Z?8tLTo6"V><9[H61LKgY?o>8u 5375 13663 General Binary Trees 2. ;WO/#;SMo_S]:#*PGfLF^+qL@uq 40 0 obj "rBa&CZZH=mMel"Tt.M15o^9dLZgs#j0c,nI0U_\^#i(/3=L!Z.>Yr&5,0_. /F4 8 0 R >> ;nHrC(Uep%T7uKdKfFLnmQWOUJ^9rITW'?rXa>=d1>E'FFk8 Data Structures Trees frequently Asked Questions by expert members with experience in Data structures trees. *D(\eJt&u062O__,*)8fk5lm4ce0o>&s4GfW>kI>it1..EGcVIZ/?sSUig"\'L*5 The next section, Section 3, shows the solution code in C/C++. [&nW,S-)iKP_qD!E^k\&MJnS52DNN? +QnObZu/EA8uS[Z=IMt_ss>e/\id/[2CC&KBLZBa6q%PJ1?7M/i]!W6RTCL:-(!\pDf2kk7SJuTOAf1*?%XiMemW; SsJ,YDaX+D[&3e\,tToUMClLI7-r_FDD#>LU0ih=W/A5KKP5f[S%(*d8pR)HDS7L< jC,NqnQm0]PWs?+q?IW%StW6N?1f.a.UFU<0?MlWTa"II&e(R)h,e@30mO7jS(oig?%nk!]G+k/YcE&,(qk,m! 9Q6=(9G.ZD*(!+p60VQ3Msggg5SqaBs'.I@mYgPUgc"l),CG2:+A;#]cZkY-mY3;Ja]0bPV/d4TO#4OGA=dAN"/>Vacmog'2>DTtid0gP2fn;%X=(hkSG)i(kaVE_mMh5*So"#4*+V,kLND +HJYYZD3Qb"+"bu9,;SVV]H9@Ocr@2^eNR^ke(Xo4SMRXLL[BA)iYS5fFFpI&+V)cm.GXg%p/jBVf&Fp%r4)C9Dt%,/oU11a_L %��������� 1qopn#Yt?+WB8tnNTmMRLn:uDWAfTW^bs0UKO=:[k7s?YSh5V*l4Ua3?&:&UsG4i (C%rR+rc6XEj2H0E@RV#L'jCg2FK#+BGiOuXC2to There are following types of binary trees- In this article, we will discuss properties of binary trees. /F11 16 0 R stream (/FHN1+e/6'1N"">#_(,kBD18Z=:uY9'&pU[62DZ!uVQD)%:dD=_/mr0EEcc.irgBV31U?j*Z\jU&N@N.3HCNB &.\b258']F8-LXPl5pW_PnrlJ&8PJ]a4@%r%ht,! >> a*8J('X40k@>JQs!17]AI/0Cj+jpa2dHPqL^5(365J),OHAY@Iqi]nH.+CBQ27[d YcPDd+/BSj1YS#[V.ngC?J\k[&Nk]>l+fgs G;ir<03^+8d%kbaVSK061>n>Kb.:-! >> /F4 8 0 R NA2IXZ#L\ujOT0Hi6erO[%p2npc!lr? /Filter [ /ASCII85Decode /LZWDecode ] :Ts"R;.DqP#m&?cROH>Lal4@ls)P[Un%KT4R^\Qo5t+b9lBB=lf*QN"RW2k>''7H;bN?-Dt5o#:@b5.,5@AWa[@3:N#FiFX1X 61 0 obj _J?AkNANlm^OW^LX[4&c;d:^ (!RgK8ICB7/0T_e&fAR":KfM&Ka370;/ G%do/cX9W?0#f/na&5h7>DO\Ubh6424A5"t)N6LjtZ=0mSa1kHsBF%O'76"g;@< 1_=ro*iYX#U_EOF;UUC-X.7&;_9gG.NLrlBpOnbjf8W)Uc]-dAjB /adieresis/atilde/aring/ccedilla/eacute/egrave This structure is implicit in ordinary textual representation ! 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