Data Structures Week #5. Trees (Ağaçlar)
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1 Data Structures Week #5 Trees Ağaçlar)
2 Trees Ağaçlar) Toros Gökarı Avrupa Gökarı October 28, 2014 Boraha Tümer, Ph.D. 2
3 Trees Ağaçlar) October 28, 2014 Boraha Tümer, Ph.D. 3
4 Outlie Trees Deiitios Implemetatio o Trees Biary Trees Tree Traversals & Expressio Trees Biary Search Trees October 28, 2014 Boraha Tümer, Ph.D. 4
5 Deiitio o a Tree Deiitio: A tree is a collectio o odes vertices) where the collectio may be empty. Otherwise, the collectio cosists o a distiguished ode r, called the root, ad zero or more o-empty sub)trees T 1,, T k, each o whose roots are coected by a directed edge to r. October 28, 2014 Boraha Tümer, Ph.D. 5
6 More deiitios I-degree o a vertex is the umber o edges arrivig at that vertex. Out-degree o a vertex is the umber o edges leavig that vertex. Nodes with out-degree=0 o childre) are called the leaves o the tree. Root is the oly ode i the tree with i-degree=0. The child C o a ode A is the ode that is coected to A by a sigle edge. The, A is the paret o C. Gradparet ad gradchild relatios ca be deied i the same maer. Nodes with the same paret are called sibligs. October 28, 2014 Boraha Tümer, Ph.D. 6
7 More deiitios A path rom a ode 1 to k is deied as a sequece o odes 1,, k such that i is the paret o i+1 or 1i<k. The legth o this path is the umber o edges o the path, i.e., k-1. There is exactly oe path rom the root to each ode. The depth o ay ode i is the legth o the uique path rom the root to i. The depth o root is 0. The height o ay ode i is the legth o the logest path rom i to a lea. October 28, 2014 Boraha Tümer, Ph.D. 7
8 A Geeral Tree Example Root A Childre o A B, C, D, E, F. Leaves o the tree C, D, G, H, I, K, L, M, ad N. Sibligs o G H ad I. October 28, 2014 Boraha Tümer, Ph.D. 8
9 Remarks The height o all leaves are 0. The height o a tree is the height o its root. The height o the above tree is 3. The height o C is 0. I there is a path rom 1 to 2, the 1 is a acestor o 2 ad 2 is a descedat o 1. October 28, 2014 Boraha Tümer, Ph.D. 9
10 Implemetatio o Trees Oe way to implemet a tree would be to have a poiter i each ode to each child, besides the data it holds. This is ieasible! Q:Why? A:The umber o childre o each ode is variable, ad hece, ukow. Aother optio would be to keep the childre i a liked list where the irst ode o the list is poited to by the poiter i the paret ode as a header. A typical tree ode declaratio i C would be as ollows: struct TNode_type { it data; struct TNodetype *childptr, *sibligptr; } October 28, 2014 Boraha Tümer, Ph.D. 10
11 A Geeral Tree Implemetatio October 28, 2014 Boraha Tümer, Ph.D. 11
12 Biary Trees BTs) A biary tree is a tree o odes with at most two childre. Declaratio o a typical ode i a biary tree struct BTNodeType { iotype *data; struct BTNodeType *let; struct BTNodeType *right; } Average depth i BTs: O ) October 28, 2014 Boraha Tümer, Ph.D. 12
13 A Biary Tree Topology October 28, 2014 Boraha Tümer, Ph.D. 13
14 Tree Traversals A tree ca be traversed recursively i three dieret ways: i-order traversal let-ode-right or LNR) First recursively visit the let subtree. Next process the data i the curret ode. Fially, recursively visit the right subtree. pre-order traversal ode-let-right or NLR) post-order traversal let-right-ode or LRN) October 28, 2014 Boraha Tümer, Ph.D. 14
15 Recursive I-order LNR) Traversal void i-orderbtnodetype *p) { i p!=ull){ i-orderp->let); operatep); i-orderp->right); } } October 28, 2014 Boraha Tümer, Ph.D. 15
16 Recursive Pre-order NLR) Traversal void pre-orderbtnodetype *p) { i p!=ull){ operatep); pre-orderp->let); pre-orderp->right); } } October 28, 2014 Boraha Tümer, Ph.D. 16
17 Recursive Post-order LRN) Traversal void post-orderbtnodetype *p) { i p!=ull){ post-orderp->let); post-orderp->right); operatep); } } October 28, 2014 Boraha Tümer, Ph.D. 17
18 Expressio Trees Preix, postix ad iix ormats o arithmetic expressios Iix Postix Preix A+B AB+ +AB A/B+C) ABC+/ /A+BC A/B+C AB/C+ +/ABC A-B*C+D/E+F) ABC*-DEF+/+ +-A*BC/D+EF A*B+C)/D-E)+F)-G/H-I) ABC+DE-/F+*GHI-/- -*A+/+BC-DEF/G-HI October 28, 2014 Boraha Tümer, Ph.D. 18
19 Costructio o a Expressio Tree rom Postix Expressios Iitialize a empty stack o subtrees Repeat Get toke; i toke is a operad Push it as a subtree else pop the last two subtrees ad orm & push a subtree such that root is the curret operator ad let ad right operads are the ormer ad latter subtrees, respectively Util the ed o arithmetic expressio October 28, 2014 Boraha Tümer, Ph.D. 19
20 Example to Costructio o Expressio Trees Empty Stack o subtrees ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 20
21 Example to Costructio o Expressio Trees - 1 Stack o subtrees A ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 21
22 Example to Costructio o Expressio Trees - 2 Stack o subtrees A B ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 22
23 Example to Costructio o Expressio Trees - 3 Stack o subtrees A B C ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 23
24 Example to Costructio o Expressio Trees - 4 Stack o subtrees A + B C ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 24
25 Example to Costructio o Expressio Trees - 5 Stack o subtrees A + D B C ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 25
26 Example to Costructio o Expressio Trees - 6 Stack o subtrees A + D E B C ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 26
27 Example to Costructio o Expressio Trees - 7 Stack o subtrees A + - B C D E ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 27
28 Example to Costructio o Expressio Trees - 8 Stack o subtrees A + / - B C D E ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 28
29 Example to Costructio o Expressio Trees - 9 Stack o subtrees A + / - F B C D E ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 29
30 Example to Costructio o Expressio Trees - 10 Stack o subtrees A + / F + - B C D E ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 30
31 Example to Costructio o Expressio Trees - 11 Stack o subtrees A * + / F + - B C D E ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 31
32 Example to Costructio o Expressio Trees - 12 Stack o subtrees A * G + / F + - B C D E ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 32
33 Example to Costructio o Expressio Trees - 13 Stack o subtrees A * G H + / F + - B C D E ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 33
34 Example to Costructio o Expressio Trees - 14 Stack o subtrees A * G H I + / F + - B C D E ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 34
35 Example to Costructio o Expressio Trees - 15 Stack o subtrees A * + G + / F H - - I B C D E ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 35
36 Example to Costructio o Expressio Trees - 16 Stack o subtrees * / A + / F G H I B C D E ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 36
37 Example to Costructio o Expressio Trees Stack o subtrees A * + / F G / B C D E ABC+DE-/F+*GHI-/- October 28, 2014 Boraha Tümer, Ph.D. 37 H I
38 Biary Search Trees BSTs) BSTs are biary trees with keys placed i each ode i such a maer that the key o a ode is greater tha all keys i its let sub-tree ad less tha all keys i its right sub-tree. Here, we assume o replicatio o keys. For replicatig keys, the relatios are modiied as greater tha or equal to or less tha or equal to depedig o the applicatio. October 28, 2014 Boraha Tümer, Ph.D. 38
39 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 39
40 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 40
41 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 41
42 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 42
43 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 43
44 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 44
45 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 45
46 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 46
47 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 47
48 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 48
49 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 49
50 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 50
51 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 51
52 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 52
53 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 53
54 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 54
55 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 55
56 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 56
57 A Biary Search Tree Example October 28, 2014 Boraha Tümer, Ph.D. 57
58 Recursive Fid Fuctio i BSTs it id BTNodeType *p, it key) { // we assume iotype i BTNodeType is it i p!= NULL) i key < p->data) idp->let, key); else i key > p->data) idp->right, key); else i p->data == key) prit p->data); } October 28, 2014 Boraha Tümer, Ph.D. 58
59 Aalysis o Fid - 1 We would like to calculate the search time, t), it takes to id a give key i a BST with odes or to id it is ot i the BST!). First let us aswer the ollowig questio: Questio: What is the uit operatio perormed to accomplish the search? Aswer: The compariso o the key searched with the key stored i the tree ode i.e., i key < p->data) or i key > p->data) or i key == p->data) ). Hece, what we eed to do to id t) is to cout, as a uctio o, how may times the compariso is executed. October 28, 2014 Boraha Tümer, Ph.D. 59
60 Aalysis o Fid - 2 Now, we have coverted the problem ito oe as i the ollowig: what is the average umber o comparisos perormed to id a give key i a BST? or how may odes, o average, are visited to id a give key or to id it is ot i the BST? or, i a higher level o detail, what is the average depth o the ode that stores the key we are lookig or or, i case the key is ot i the BST, what is the average depth o the lea at the ed o the path our search ollows through? October 28, 2014 Boraha Tümer, Ph.D. 60
61 Aalysis o Fid - 3 Now, we tur back to ormulatig t): We will do the ormulatio or three cases: 1. Average case 2. Worst case 3. Best case October 28, 2014 Boraha Tümer, Ph.D. 61
62 Aalysis o Fid: Average Case - 1 Average case: Each ode has a let ad right subtree. The search time will cosist o the search time o the let subtree, right subtree ad the time spet or the root i the mai BST. That is, the sum o all depths o odes i a BST also kow as the iteral path legth) will be the sum o the depths o those odes i the let subtree, those i the right subtree ad the additioal cotributio o the root i the mai tree. We will ormulate ad id the iteral path legth ) o a BST with odes. Assume the let ad right subtrees o the root o a BST with odes have k ad -k-1 odes, respectively. The the depth may be expressed as ollows: )=k)+-k-1)+-1; The term -1 is added to the sum o depths due to the act that all odes i the BST are oe level deeper i the mai BST because o the root o the mai BST). October 28, 2014 Boraha Tümer, Ph.D. 62
63 A example tree with 14) k) ) -k-1) 14) 14) 14) t ) 3) 10) ) t ) October 28, 2014 Boraha Tümer, Ph.D. 63
64 Aother example tree with 14) -k-1) k) ) 14) 9) 4) ) ) 36 t ) ) t ) October 28, 2014 Boraha Tümer, Ph.D. 64
65 Average Case - 2 )=k)+-k-1)+-1; k ad -k-1 ca be ay umber betwee betwee 0 ad -1. To come up with a geeral solutio, we ca average k) ad -k-1): ave ) 2 1 k0 k) 1 From this poit o we will drop the subscript o ad use ) to deote ave ) October 28, 2014 Boraha Tümer, Ph.D. 65
66 October 28, 2014 Boraha Tümer, Ph.D. 66 Average Case - 3 1) 1) 2 1) 1 ) 1); 2 1) 1) ) ) ) 1); 2 1) 2 1) 1) ) ) 2) 1) ) 2 1) 1) ) 1); ) 2 ) 1* ) 2 ) II I II k I k k k k k
67 October 28, 2014 Boraha Tümer, Ph.D. 67 Average Case - 4 i i i i i i i i 1 1 1) 1 1) 2 0) 1) ) 1) 1 2 0) 1 ) 0 1 0) 2 1) 2* ) 3 2) 1) ) 1) 1) 1 2 1) 1 )
68 October 28, 2014 Boraha Tümer, Ph.D. 68 Average Case - 5 )) log )) log ) log ; 1 1) 2 4 1) 2 ) ) 2 ) ) 2 ) 0 0) ; ) 1 ; 1) 1 1) 2 0) 1) ) ) log O O c x a x dx i i i i i i i i i i i i i i i i i To id the average per ode) umber o comparisos, N ave, we divide ) by the umber o odes : N ave = )/Olog )
69 Aalysis o Fid: Worst Case - 1 Worst Case BST: The worst case BST is oe with the deepest path. A - ode such BST is oe with depth levels. Questio: How may such -ode BSTs are there? We will ormulate ad id the iteral path legth ) i a worst case -ode BST. I such a BST oe subtree o the root orms a liked list o -1 odes whereas the other subtree has o odes. The the depth may be expressed as ollows: )=-1)+-1. October 28, 2014 Boraha Tümer, Ph.D. 69
70 October 28, 2014 Boraha Tümer, Ph.D. 70 Worst Case - 2?) ) ) ) 0 1) ) 0 1) 1) : 0 1) 1; 1) ) O c c c c c c c c c c c c x x CE Order o ): )=O 2 )
71 Aalysis o Fid: Best Case - 1 Best Case BST: The best case BST is oe with the least tree depth. A -ode such BST is oe with log 2 1 depth levels. Questio: How may such -ode BSTs are there? We will ormulate ad id the iteral path legth ) i a best case -ode BST. I such a BST both subtrees o the root have the same umber o odes i.e., /2 odes). The the depth may be expressed as ollows: )=2/2)+-1. October 28, 2014 Boraha Tümer, Ph.D. 71
72 October 28, 2014 Boraha Tümer, Ph.D. 72 Best Case - 2?) ) ) log 2 1 ) ) 0 1) ) log ) 2 2 ) 0 1) 2) 2) : 1 2 1) 2 ) 2 0; 1) 1; 2) / 2 ) O c c c c c c k c c k x x x CE k k k k k k Order o ): )=Olg)
73 No-recursive FidMi Fuctio i BSTs BTNodeType *idmi BTNodeType *p) {// returs a poiter to ode with the miimum key i the BST. // which oe is the ode with the miimum key i a BST? i p!= NULL) while p->let!= NULL) p=p->let; retur p; } October 28, 2014 Boraha Tümer, Ph.D. 73
74 Recursive FidMax Fuctio i BSTs BTNodeType *idmax BTNodeType *p) {// returs a poiter to ode with the maximum key i the BST // which oe is the maximum key i a BST? i p == NULL) retur NULL; i p->right == NULL) retur p; retur idmaxp->right); } October 28, 2014 Boraha Tümer, Ph.D. 74
75 Recursive Isert Fuctio i BSTs BTNodeType *Isert iotype * key, BTNodeType *p) {// returs a poiter to ew ode with key iserted i the BST. } i p == NULL) { //empty tree p=mallocsizeobtnodetype)); i p == NULL) retur OutoMemoryError; //o heap space let!!! p->data=key; p->let=p->right=null; retur p; } else i key < p->data) //tree exists, key < root p->let=isertkey,p->let); else i key > p->data) //tree exists, key > root p->right=isertkey,p->right); // retur p; //key oud October 28, 2014 Boraha Tümer, Ph.D. 75
76 Recursive Remove Fuctio i BSTs BTNodeType *Remove iotype *key, BTNodeType *p) {// returs a poiter to ode replacig the ode removed. BTNodeType *TempPtr; i p == NULL) retur errormessage empty tree ); i key < p->data) p->let=removekey,p->let); else i key > p->data) p->right=removekey,p->right); else i p->right && p->let) { TmpPtr=idMip->right); p->data=tmpptr->data; p->right=removep->data,p->right); } } else { TmpPtr=p; i p->let == NULL) p=p->right; else i p->right == NULL) p=p->let; reetmpptr); } retur p; //empty tree //ode with key oud!!! //i ode has two childre //id smallest key o right subtree o ode with key //replace removed data by smallest key o right subtree //i ode has less tha two childre October 28, 2014 Boraha Tümer, Ph.D. 76
77 Example or Remove Fuctio...1 Remove i p->right && p->let) {// two childre TmpPtr=idMip->right); //id smallest key //o right subtree p->data=tmpptr->data; // traser key p->right=removep->data,p->right); //replace } p Miimum o Right subtree October 28, 2014 Boraha Tümer, Ph.D. 77
78 Example or Remove Fuctio...1 Remove i p->right && p->let) {// two childre TmpPtr=idMip->right); //id smallest key //o right subtree p->data=tmpptr->data; // traser key p->right=removep->data,p->right); //replace } p TmpPtr October 28, 2014 Boraha Tümer, Ph.D. 78
79 Example or Remove Fuctio...1 Remove i p->right && p->let) {// two childre TmpPtr=idMip->right); //id smallest key //o right subtree p->data=tmpptr->data; // traser key p->right=removep->data,p->right); //replace } p TmpPtr October 28, 2014 Boraha Tümer, Ph.D. 79
80 Example or Remove Fuctio...1 Remove i p->right && p->let) {// two childre TmpPtr=idMip->right); //id smallest key //o right subtree p->data=tmpptr->data; // traser key p->right=removep->data,p->right); //replace } p October 28, 2014 Boraha Tümer, Ph.D. 80
81 Best-Average-Worst Case Access Times o a speciic BST To id a ode with some speciic piece o data, a search gets started rom the root. Data at each ode is compared with the key that is searched or. Coutig up the umber o comparisos or each ode will reder the access time. October 28, 2014 Boraha Tümer, Ph.D. 81
82 Calculatio o Access Time Access to some speciic piece o data is achieved whe the ode this piece o data resides is oud. To id this ode, a search gets started rom the root. Data at each ode is compared with the key that is searched or. Coutig up the umber o comparisos or each ode will reder the access time. October 28, 2014 Boraha Tümer, Ph.D. 82
83 Calculatio o Access Time Depth level # o Comparisos/ ode # o Comparisos/ level 0 1 1*1= *2= *3= *5= *7= *1=6 Total Nodes: BST Access Times Average 75/19=3.94 Worst 6 Best 1 October 28, 2014 Boraha Tümer, Ph.D. 83
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