Announcements TREES II. Comparing Data Structures. Binary Search Trees. Red-Black Trees. Red-Black Trees 3/13/18

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1 //8 Aoucemets Prelim is Toight, brig your studet ID :PM EXAM OLH: etids startig aa to dh OLH: etids startig di to ji PHL: etids startig jj to ks (Plus studets who switched from the 7: exam) TREES II Lecture CS Sprig 8 7:PM EXAM ( Studets) OLH: etids startig kt to rz OLH: etids startig sa to wl PHL: etids startig wm to zz (Plus studets who switched from the : exam) Comparig Data Structures Biary Search Trees Data Structure add(val x) lookup(it i) Array Liked List O() Biary Tree O() BST O() search(val x) O(height) O(height) O(height) jauary february march april jue may august july september april august december february december ovember october jauary RedBlack Trees Selfbalacig BST Each ode has oe extra bit of iformatio "color" Costraits o how odes ca be colored eforces approximate balace RedBlack Trees ) A redblack tree is a biary search tree. ) Every ode is either red or black. ) The root is black. ) If a ode is red, the its (oull) childre are black. ) For each ode, every path to a decedat ull ode cotais the same umber of black odes.

2 //8 RB Tree Quiz Class for a RBNode 7 8 Which of the followig are redblack trees? A) B) C) D) class RBNode<T> { private T value; private Color color; private RBNode<T> paret; private RBNode<T> left, right; Null if the ode is the root of the tree. Either might be ull if the subtree is empty. / Costructor: oeode tree with value x / public RBNode (T v, Color c) { value= d; color= c;... YES NO YES NO Isert fixtree 9 Isert(RBTree t, it v){ Node p; Node = t.root; while(!= ull){ p= ; if(v <.value){=.left else{=.right Node vode= ew Node(v, RED) if(p == NULL){ t.root= vode; else if(v < p.value){ p.left= vode; vode.paret= p; else{ p.right= vode; vode.paret= p; fixtree(t, vode); p 8 Case : paret is black Case : paret is red ucle is black ode o outside Case : paret is red ucle is black ode o iside Case : paret is red ucle is red Rotatios fixtree p leftrotate rightrotate p fixtree(rbtree t, RBNode ){ while(.paret.color == RED){ // ot Case if(.paret.paret.right ==.paret){ Node ucle =.paret.paret.left; if(ucle.color == BLACK) { // Case or if(.paret.left == ) { rightrotate(); //.paret.color== BLACK;.paret.paret.color= RED; leftrotate(.paret.paret); else { //ucle.color == RED // Case.paret.color= BLACK; ucle.color= BLACK;.paret.paret.color= RED; =.paret.paret; else {... //.paret.paret.left ==.paret t.root.color == BLACK;// fix root

3 //8 Search Redblack trees are a special case of biary search trees Search works exactly the same as i ay BST Time: O(height) Observatio : Every biary tree must have a ull ode with depth log Observatio : Every biary tree must have a ull ode with depth log log() Observatio : Every biary tree must have a ull ode with depth log Observatio : I a redblack tree, the umber of red odes i a path from the root to a ull ode is less tha or equal to the umber of black odes. 7 8 Observatio : Every biary tree must have a ull ode with depth log Observatio : I a redblack tree, the umber of red odes i a path from the root to a ull ode is less tha or equal to the umber of black odes. Observatio : The maximum path legth from the root to a ull ode is at most times the miimum path legth from the root to a ull ode. Observatio : Every biary tree must have a ull ode with depth log Observatio : I a redblack tree, the umber of red odes i a path from the root to a ull ode is less tha or equal to the umber of black odes. Observatio : The maximum path legth from the root to a ull ode is at most times the miimum path legth from the root to a ull ode. h = max path le mi path le log ( ) 89:: 89:: h is O(log )

4 //8 Comparig Data Structures Applicatio of Trees: Sytax Trees 9 Data Structure add(val x) lookup(it i) Array Liked List O() Biary Tree O() BST RB Tree O() search(val x) O(height) O(height) O(height) O(log ) O(log ) O(log ) Most laguages (atural ad computer) have a recursive, hierarchical structure This structure is implicit i ordiary textual represetatio Recursive structure ca be made explicit by represetig seteces i the laguage as trees: Abstract Sytax Trees (ASTs) ASTs are easier to optimize, geerate code from, etc. tha textual represetatio A parser coverts textual represetatios to AST Applicatios of Trees: Sytax Trees Preorder, Postorder, ad Iorder parsig ( ) A Java expressio as a strig. Preorder traversal:. Visit the root. Visit the left subtree (i preorder). Visit the right subtree A expressio as a tree. Preorder, Postorder, ad Iorder Preorder, Postorder, ad Iorder Preorder traversal Preorder traversal Postorder traversal. Visit the left subtree (i postorder). Visit the right subtree. Visit the root Postorder traversal Iorder traversal. Visit the left subtree (iorder). Visit the root. Visit the right subtree

5 //8 Preorder, Postorder, ad Iorder Pritig cotets of BST (IOrder Traversal) Preorder traversal Postorder traversal Iorder traversal ( ) ( ) To avoid ambiguity, add paretheses aroud subtrees that cotai operators. Because of orderig rules for a BST, it s easy to prit the items i alphabetical order Recursively prit left subtree Prit the ode Recursively prit right subtree / Prit BST t i alpha order / private static void prit(treenode<t> t) { if (t== ull) retur; prit(t.left); System.out.prit(t.value); prit(t.right); 7 8 9

6 //8 I about 97, Gries paid $ for a HP calculator, which had some memory ad used postfix otatio! Still works. a.k.a. reverse Polish otatio I Defese of Prefix Notatio Iterator/Iterable Fuctio calls i most programmig laguages use prefix otatio: like add(7, ). Some laguages (Lisp, Scheme, Racket) use prefix otatio for everythig to make the sytax simpler. (defie (fib ) (if (<= ) ( (fib ( ) (fib ( ))))) There's a pair of Java iterfaces desiged to make data structures easy to traverse You could modify a tree to implemet iterable, implemet a (iorder, postorder, etc.) iterator ad the use a for each loop to traverse the tree! I recitatio this week, you will modify your liked list from A to implemet iterable