Review: Binary Trees. CSCI 262 Data Structures. Search Trees. In Order Traversal. Binary Search Trees 4/10/2018. Review: Binary Tree Implementation

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1 Rviw: Binry Trs CSCI 262 Dt Struturs 21 Binry Srh Trs A inry tr is in rursivly: = or A inry tr is (mpty) root no with lt hil n riht hil, h o whih is inry tr. Rviw: Binry Tr Implmnttion Just ollow th rursiv inition to t simpl implmnttion: lss inry_tr_no { puli: T t; inry_tr_no<t>* lt; inry_tr_no<t>* riht; Srh Trs Dt strutur or holin omprl lmnts Eiint srhin, insrtion, ltion Unrlyin strutur or sts, mps (BSTs) Also us in ts inxin (B-Trs) Th si strutur: Nos hol uniqu t vlus n pointrs to hil nos Dt ts s prtitionin lmnt in th tr: Chil nos/trs to th lt o th t lmnt hv vlu lss thn th t lmnt Chil nos/trs to th riht hv vlu rtr thn th t lmnt Binry Srh Trs In Orr Trvrsl Hr s inry srh tr (BST): Nos ontin strins Lt hil sutr ontins only vlus lss thn root Root vlu is lss thn ll riht hil sutr vlus Visit lt sutr, visit root, visit riht sutr. voi print_inorr(inry_tr_no<t>* root) { i (root!= NULL) { print_inorr(root->lt); out << root->t << ; print_inorr(root->riht); ppl nn hrry uv lmon orn ph pr quin 1

2 Srhin Insrtin Exmpl: srh (root, ) Whr o w insrt n itm into th tr? How woul you to this tr? inry_tr_no<t>* srh(inry_tr_no<t>* root, T vl) { i (root == NULL) rturn NULL; i (vl == root->t) rturn root; i (vl < root->t) rturn srh(root->lt, vl); ls rturn srh(root->riht, vl); Answr: put it whr you xpt to in it! Insrtin Rmovin voi insrt(inry_tr_no<t>*& root, T vl) { i (root == NULL) root = nw inry_tr_no<t>(vl); ls i (vl < tr->ino) insrt(tr->lt, vl); ls i (vl > tr->ino) insrt(tr->riht, vl); Exmpl: insrt(root, ) voi rmov(inry_tr_no<t>* root, T vl) { // this is trikir!... Wht i ws lry in our tr? Wht o w o irst? Wht is th s s? Rmovin Rmovin Cs 1: No Chil 3 Css whn no is in tr: 1. No hilrn 2. On hil 3. Two hilrn 1. Fin th itm 2. Dth n lt Exmpl: rmov(root, ) 2

3 Rmovin Cs 2: On Chil Rmovin Cs 3: Two Chilrn 1. Fin th itm 2. Link hil to prnt 3. Dlt Exmpl: rmov(root, ) 1. Fin th itm 2. Swp with rihtmost itm in lt sutr (why?) 3. Rmov rihtmost no in lt sutr (Cs 1 or 2) Exmpl: rmov(root, ) Rmovin: Co Prti With BSTs voi rmov(inry_tr_no<t>*& root, T vl) { i (root == NULL) rturn NULL; i (vl < root->t) rmov(root->lt, vl); ls i (vl > root->t) rmov(root->riht, vl); ls { // itm oun! i (root->lt == NULL root->riht == NULL) { inry_tr_no<t>* tmp; i (root->lt == NULL) tmp = root->riht; ls tmp = root->lt; lt root; root = tmp; ls { inry_tr_no<t>* tmp = root->lt; // in rihtmost no inry_tr_no<t>* prnt = root; // in lt sutr whil (tmp->riht!= NULL) { prnt = tmp; tmp = tmp->riht; root->t = tmp->t; // opy t to root i (prnt == root) // th n lt rihtmost no root->lt = tmp->lt; // in lt sutr ls prnt->riht = tmp->lt; lt tmp; Bs s: itm not oun, o nothin. Rursiv lls to in itm to lt. Css 1 n 2, oth. Cn you s why s 1 is hnl y this lok? Cs Anlysis Complxity o Srh Wht is th i-o omplxity o: Srhin? Insrtin? Rmovin? srh(root, ) How mny rursiv stps? A. <= hiht o tr 3

4 Hiht o Trs So how hih is tr with N nos? Hiht-Bln Trs (AVL) Ain, rursiv inition: Lt n riht sutrs r hiht-ln Lt n riht sutrs ir in hiht y no mor thn 1 Bst s: h = lo 2 (N+1) = O(lo N) Worst s: h = N Whih o ths r hiht ln? h Anlysis on Bln BSTs Whn trs r ln: Eh sutr ontins rouhly hl th nos Eh stp own th tr rouhly hlvs th prolm inry_tr_no<t>* srh(inry_tr_no<t>* root, T vl) { i (root == NULL) rturn NULL; i (vl == root->t) rturn root; i (vl < root->t) rturn srh(root->lt, vl); rturn srh(root->riht, vl); Srh, insrt, lt r ll O(lo N) Sl Blnin BSTs Trs om unln throuh sris o insrts n lts Sl-lnin: prorm O(lo N) or wr oprtions to rln tr insrt, lt Exmpls o sl-lnin BSTs: R-Blk trs AVL trs Sply trs Rottions Rlnin Exmpl (AVL) W hn th ln t no vi rottion: y x x y C A A B This is th riht rottion. Th lt rottion is th mirror im o this on. B C Rmovin this no unlns th tr t. A sinl riht rottion rstors ln. I th tr shown ws BST, is th nw tr BST? 4

5 Tr Blnin (AVL) Finl Wors Why BSTs mttr: Linux krnl: shulrs, xt3 ilsystm, virtul mmory, mny mor (R-Blk trs) Orr st n mp typs (.., C++ STL, Jv) (R-Blk trs in!) Dts inxin (B-trs not xtly BSTs, ut rlt) Filsystm mtt inxin (B-trs or R-B) Lurkin in your vorit OS? Up Nxt Friy, April 13 E-DAYS NO CLASS Mony, April 16 Mitrm rviw APT 5 u Wnsy, April 18 Mitrm 2 (in lss) Friy, April 20 Fun & Gms inst o l (optionl) 5