CSC 143S 6 CSC 143S 3

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1 CSC 143 Trees [Chpter 10] Everydy Exmples of Trees Fmily tree Compny orgniztion chrt Tble of contents Any hierrchicl orgniztion of informtion, representing components in terms of their subcomponents NB: Some of these re not true trees in the CS definition of the term CSC 143S 1 CSC 143S 2 A Tree d edges b leves nodes c g k m e f h i l CSC 143S 3 Tree Terminology Empty tree: tree with no nodes Child of node u Any node rechble from u by 1 edge pointing wy from u Nodes cn hve zero, one, or more children Leves hve no children If b is child of, then is the prent of b All nodes except hve exctly one prent Root hs no prent CSC 143S 4 Tree Terminology (2) Descendnt of node (recursive definition) P is descendnt of P If C is child of P, nd P is descendnt of A, then C is descendnt of A Exmple: k nd m re descendnts of, l is descendnt of, k, nd l Ancestor of node If D is descendnt of A, then A is n ncestor of D Exmple:, k, nd l re k l m Tree Terminology (3) Subtree Any node of tree, with ll of its descendnts Depth (recursive definition) Depth of node is 0 Depth of ny node other thn is one greter thn depth of its prent ncestors of l CSC 143S 5 CSC 143S 6 c c b b g g k m k m

2 Tree Terminology (4) Height Height of tree is mximum of ll depths of its leves Wrning: Definitions vry Some define depth of the node s 1. Binry Trees A binry tree is tree ech of whose nodes hs exctly zero, one, or two children Two children re clled the left child nd right child Left child b i Right child c f d e g h k CSC 143S 7 CSC 143S 8 Binry Tree Dt Structure Usully use vrint of Node: struct BTreeNode { int dt; BTreeNode *left; BTreeNode *right; ; dt left Keep pointer to the node Empty tree hs NULL Note the recursive dt structure As the dt structure is recursive, lgorithms often re recursive s well right CSC 143S Exmple: Counting Nodes Bse cse: Empty tree hs zero nodes Recursive cse: Nonempty tree hs + nodes in left subtree + nodes in right subtree int CountNodes(BTreeNode *) { if ( == NULL ) return 0; // bse return 1+CountNodes(->left) +CountNodes(->right); CSC 143S 10 Finding the Height Bse cse: Empty tree hs height -1 Recursive cse: Nonempty tree hs height 1 more thn mximum height of left nd right subtrees int Height(BTreeNode *) { if ( == NULL ) return -1; return 1+mx(Height(->left), Height(->right)); Anlyses Wht is running time of these lgorithms? Bse cse: O(1) for both Recursive cse: O(N) for both, where N is the number of nodes in tree How to write n itertive version? Try it CSC 143S 11 CSC 143S

3 Recursive Tree Serching How to tell if dt item is in binry tree? bool Find(BTreeNode *,int item){ if ( == NULL ) return flse; if ( ->dt == item ) return true; return (Find(->left,item) Find(->right,item)); CSC 143S 13 Recursive Tree Serching (2) Wht is the running time of this lgorithm? Worst cse: Hs to visit every node in the tree, O(N) Cn we do better? CSC 143S 14 Binry Serch Trees [Section 13.3] Properties: No duplictes Prereq: The greter-thn nd less-thn reltions re well-defined for the dt vlues. Sorting constrints: for every node v All dt in left subtree of v < dt in All dt in right subtree of v > dt in A binry tree with these constrints is clled binry serch tree (BST) Given set of vlues, there could be mny Exmples nd Non-Exmples A Binry Serch Tree Not Binry Serch Tree possible BSTs CSC 143S 15 CSC 143S 16 Finding n item in BST Find(,6) Find(,10) 4 NULL CSC 143S 17 Finding n item in BST (2) If we hve binry serch tree, then Find cn be done s: bool Find(BTreeNode *, int item) { if ( == NULL ) return flse; if (item == ->dt) return true; if (item < ->dt) return Find(->left, item); return Find(->right, item); CSC 143S 18

4 Running time of Find Best cse: O(1), item is t Worst cse: O(h), where h is height of tree Leds to question: Wht is the height of binry serch tree with N nodes? Perfect tree (2 d nodes t ech depth d) is best cse: N = 2 h+1-1 h = log 2 (N+1) - 1 = O(log N) logrithmic running time CSC 143S 1 Running time of Find (2) Wht if tree isn t blnced? Worst cse is degenerte tree Height = N-1, N is number of nodes Running time of Find, worst-cse, is O(N) CSC 143S 20 Tbles Revisited Using List or Arry to implement Tble cn be inefficient for serching Could use binry serch trees to implement tble (dictionry) Must support Insert nd Delete in ddition to Find Must mintin BST ordering constrint Inserting in BST To insert new key: Bse cse: If tree is empty, crete new node for item If holds key, return (no duplicte keys llowed) Recursive cse: If key < s vlue, recursively dd to left subtree, otherwise to right subtree CSC 143S 21 CSC 143S 22 Exmple Add 8, 10, 5, 1, 7, 11 to n empty BST, in tht order: CSC 143S 23 Inserting in BST (2) //Add dt to tree nd return ptr to tree BTreeNode* Insert(BTreeNode*, int dt){ if ( == NULL ) { BTreeNode *tmp = new BTreeNode; tmp->left = tmp->right = NULL; tmp->dt = dt; return tmp; if (dt < ->dt ) ->left = Insert(->left, dt); if (dt > ->dt ) ->right = Insert(->right, dt); return ; CSC 143S 24

5 Exmple (2) Wht if we chnge the order in which the numbers re dded? Add 1, 5, 7, 8, 10, 11 to BST, in tht order (following the lgorithm): CSC 143S 25 Complexity of Insert Bse cse: O(1) How mny recursive clls? For ech node dded, tkes O(H), where H is the height of the tree Agin, wht is height of tree? Blnced trees yields best-cse height of O(log N) for N nodes Degenerte trees yield worst-cse height of O(N) for N nodes For rndom insertions, expected height is O(log N) -- true, but not simple to prove CSC 143S 26 Deleting n Item from BST Simple strtegy: lzy deletion (ust mrk the node s deleted ) The hrd wy. Must del with 3 cses 1. The deleted item hs no children (esy) 2. The deleted item hs 1 child (hrder) 3. The deleted item hs 2 children (hrdest) CSC 143S 27 Deletion (2): Algorithm First find the node (cll it N) to delete. If N is lef, ust delete it. If N hs ust one child, hve N s prent bypss it nd point to N s child. If N hs two children: Replce N s key with the smllest key K of the right subtree (Recursively) delete the node tht hd key K (this node is now useless) Note: The smllest key lwys lives t the leftmost corner of subtree (why?) CSC 143S 28 Deletion (3): Finding the Node This is the esy prt: BTreeNode* delitem (int item, BTreeNode* t) { if (t!= NULL) { if (item == t->dt) t = delnode(t); if (item > t->dt) t->right=delitem(item,t->right); t->left = delitem(item,t->left); return t; CSC 143S 2 Deletion (4): Deleting the Node BTreeNode* delnode(btreenode* t) { if (t->left && t->right) { // 2 children t->dt = findmin(t->right); t->right = delitem(t->dt, t->right); return t; { // 0 or 1 child BTreeNode* rvl = NULL; if (t->left) // left child only rvl = t->left; if (t->right)// right child only rvl = t->right; delete t; return rvl; CSC 143S 30

6 Deletion (5): Finding Min All tht remins is to figure out how to find the minimum vlue in BST Remember, the minimum element lives t the leftmost corner of BST //PRECONDITION: must be clled on //non-null pointer int findmin(btreenode* t){ ssert(t!= NULL); while (t->left!= NULL) t = t->left; return t->dt; CSC 143S 31 Blnced Serch Trees BST opertions re dependent on tree height O(log N) for N nodes if tree is blnced O(N) if tree is not Cn we ensure tree is lwys blnced? Yes: Insert nd Delete cn be modified to reorgnize tree if it gets unblnced Exct detils more complicted Results in O(log N) dictionry opertions, even in worst cse CSC 143S 32 Tree Trversl Would like to be ble to iterte through ll nodes in tree How to hve Strt(), IsEnd(), NextElement()? Wht order should nodes be visited in? Top-down? Left-to-right? Bottom-up? How to del with recursive nture of trees? Visiting nodes of tree is clled tree trversl Types of Trversl Preorder trversl: Visit the node first Recursively do preorder trversl on its children, in some order Postorder trversl: Recursively do postorder trversls of children, in some order Visit the node lst CSC 143S 33 CSC 143S 34 Inorder (for Binry Trees) With binry tree, ech node hs t most two children, clled left nd right Inorder trversl: Recursively do inorder trversl of left child Then visit the node Then recursively do inorder trversl of right child For preorder nd postorder trversls, typiclly trverse left child before right one Exmple of Tree Trversl Preorder: Inorder: Postorder: l1 CSC 143S 35 CSC 143S 36

7 Another Exmple Wht bout this tree? l3 11 Preorder: Inorder: Postorder: Exmple Trversl: Printing // print the nodes of tree with // given void Preorder (BTreeNode *){ if ( == NULL ) return; // Visit the node first cout << ->dt << " "; Preorder (->left); Preorder (->right); CSC 143S 37 CSC 143S 38 Exmple Trversl: Printing void printinorder(btreenode* t){ if (t!= NULL) { printinorder(t->left); cout << t->dt << ; printinorder(t->right); void printpreorder(btreenode* t){ if (t!= NULL) { cout << t->dt << " "; printinorder(t->left); printinorder(t->right); CSC 143S 3 Another Exmple Use postorder trversl to return whole tree to the hep. void deletetree(btreenode* t) { if (t!= NULL) { deletetree(t->left); deletetree(t->right); delete t; Would inorder or preorder work s well? CSC 143S 40 Anlysis of Tree Trversl How mny recursive clls? Two for every node in tree (plus one initil cll); O(N) in totl for N nodes How much time per cll? Depends on complexity O(V) of the visit For printing nd most other types of trversl, visit is O(1)time Multiply to get totl O(N)*O(V) = O(N*V) Does tree shpe mtter? CSC 143S 41 Expression Trees [Section 13.5] Progrms hve hierrchicl structure All sttements hve fixed form Sttements cn be ordered nd nested lmost rbitrrily (nested ifthen-) Cn use structure known s syntx tree to represent progrms Trees cpture hierrchicl structure CSC 143S 42

8 A Syntx Tree Consider the C++ sttement: if ( == b + 1 ) x = y;... if expression vr equlity vr + const b 1 sttement ( expression ) sttement sttement == expression LHS = vr x expression expression vr y ;... Syntx Trees Compilers usully use syntx trees when compiling progrms Cn pply simple rules to check progrm for syntx errors (the grmmr in Appendix K of textbook) Esier for compiler to trnslte nd optimize thn text file Process of building syntx tree is clled prsing CSC 143S 43 CSC 143S 44 Binry Expression Trees A binry expression tree is syntx tree used to represent mening of mthemticl expression Norml mthemticl opertors like +, -, *, / Structure of tree defines result Esy to evlute expressions from their binry expression tree Exmple 5 * 3 + ( - 1) / / * CSC 143S 45 CSC 143S 46 Trversing Expression Trees Trverse in preorder for prefix nottion - + * 5 3 / Trverse in inorder for infix nottion 5 * / 4-1 Note tht opertor precedence my be wrong without dding prentheses t every step (((5*3) + (( - 1) / 4)) - 1) Trverse in postorder for postfix nottion 5 3 * 1-4 / CSC 143S 47 Evluting Expressions Esy to evlute n expression from its binry expression tree Use postorder trversl Recursively evlute the left nd right subtrees nd store those vlues Apply opertor t to stored vlues Much like using stck to evlute postfix nottion CSC 143S 48

9 Trees Summry Tree s new hierrchicl dt structure Recursive definition nd recursive dt structure Tree prts nd terminology Mde up of nodes Root node, lef nodes Children, prents, ncestors, descendnts Depth of node, height of tree Trees Summry (2) Binry Trees Either 0, 1, or 2 children t ny node Recursive functions to mnipulte them Binry Serch Trees Binry Trees with ordering invrint Recursive BST serch Recursive Insert, Delete functions O(H) opertions, where H is height of tree O(log N) for N nodes in blnced cse O(N) in worst cse Subtrees CSC 143S 4 CSC 143S 50 Trees Summry (3) Tree Trversls Preorder trversl Postorder trversl Binry Tree Trversls Inorder trversl Expression nd Syntx Trees CSC 143S 51