1. DATA STRUCTURES AND ALGORITHMS Lecture Notes 5 Prepared by İnanç TAHRALI

2. 2 ROAD MAP TREES Definitions and Terminology Implementation of Trees Tree Traversals Binary Trees The Search Tree ADT-Binary Search Trees AVL Trees

3. 3 TREES: Definition Recursive Definition : Tree is a collection of nodes A tree can be empty A tree contains zero or more subtrees T 1, T 2,… T k connected to a root node by edges

4. 4 TREES: Terminology Family Tree Terminology child  F is child of A parent  A is the parent of F each node is connected to a parent except the root sibling  nodes with same parents (K, L, M) leaf  nodes with no children (P, Q) Ancestor / Descendant

5. 5 Path : a sequence of nodes n 1, n 2, … n k, where n i is the parent of n i +11≤i<k Lenght : number of edges on the path (k-1) Depth : depth of n i is the lenght of unique path from the root to n i –depth of root is 0 –depth of a tree = depth of the deepest leaf Height : height of n i is the lenght of the longest path from n i to a leaf –height of a leaf is 0 –height of a tree = height of the root = depth of the tree

6. 6 Implementation of Trees 1. Each node contains a pointer to each of its children number of children would be large 2. Each node contains an array of pointers to its children number of children may vary greatly (waste of space)

7. 7 Implementation of Trees 3. Each node contains a linked list of the children struct TreeNode { Object element; TreeNode *firstChild; TreeNode *nextSibling; }

8. 8 ROAD MAP TREES Implementation of Trees Tree Traversals Binary Trees The Search Tree ADT-Binary Search Trees AVL Trees

9. 9 Tree Traversals One of the most popular applications of trees is the directory structure of O/S

10. 10 Preorder Tree Traversals work on the node first before its children ! void FileSystem::listAll(int depth = 0)const{ printName(depth); if (isDirectory()) for each file c in this directory (for each child) c.listAll(depth+1); } Pseudocode to list a directory in a hierarchical file system

11. 11 Postorder Tree Traversals work on the node after its children ! Pseudocode to calculate the size of a directory int FileSystem::size () const{ int totalSize = sizeOfThisFile (); if (isDirectory()) for each file c in this directory totalSize += c.size(); return totalsize; }

12. 12 ROAD MAP TREES Implementation of Trees Tree Traversals Binary Trees The Search Tree ADT-Binary Search Trees AVL Trees

13. 13 Binary Trees Definition : A tree in which nodes have at most two children Generic Binary Tree

14. 14 Binary Trees Binary TreeWorst case binary tree

15. 15 Implementation of Binary Trees keep a pointer to left child and right child struct BinaryNode { Object element; BinaryNode *left; BinaryNode *right; }

16. 16 Example: Expression Trees Leaves contain operands Internal nodes contain operators Inorder traversal => infix expression Preorder traversal => prefix expression Postorder traversal => postfix expression

17. 17 Expression Trees Entire tree represents (a+(b*c))+(((d*e)+f)*g) Left subtree represents a+(b*c) Right subtree represents ((d*e)+f)*g

18. 18 Constructing an Expression Tree Algorithm to convert a postfix expresion into an expression tree read the expression one symbol at a time. if the symbol is operand create a one-node tree push the tree onto a stack if the symbol is operator pop two trees T 1 and T 2 from the stack form a new tree whose root is the operator and whose left and right subtrees are T 2 and T 1 respectively This new tree is pushed onto the stack

19. 19 Example: input is ab+cde+** First two symbols are operands create one-node trees push pointers to them onto a stack Next + is read pointers to trees are poped a new tree is formed a pointer to it is pushed onto the stack

20. 20 Example: input is ab+cde+** c, d, e are read for each a one-node tree is created A pointer to the corresponding tree is pushed onto the stack.

21. 21 Example: input is ab+cde+** + is read two trees are merged

22. 22 Example: input is ab+cde+** * is read pop two tree pointers and form a new tree with a * as root

23. 23 Example: input is ab+cde+** Last symbol is read two trees are merged a pointer to the final tree is left on the stack

24. 24 ROAD MAP TREES Implementation of Trees Tree Traversals Binary Trees The Search Tree ADT-Binary SearchTrees AVL Trees

25. 25 Binary Search Trees Each node in tree stores an item items are integers and distinct deal with dublicates later Properties A binary tree for each node x values at left subtree are smaller values at right subtree are larger than the value of x

26. 26 Binary Search Trees Which one of the trees below is binary search tree ?

27. 27 Binary Search Trees template class BinarySearchTree template class BinaryNode { Comparable element; BinaryNode *left; BinaryNode *right; BinaryNode( const Comparable & theElement, BinaryNode *lt, BinaryNode *rt ) : element( theElement ), left( lt ), right( rt ) { } friend class BinarySearchTree };

28. 28 template class BinarySearchTree { public: explicit BinarySearchTree( const Comparable & notFound); BinarySearchTree( const BinarySearchTree & rhs ); ~BinarySearchTree( ); const Comparable & findMin( ) const; const Comparable & findMax( ) const; const Comparable & find (const Comparable & x) const; bool isEmpty( ) const; void printTree( ) const; void makeEmpty( ); void insert( const Comparable & x ); void remove( const Comparable & x ); const BinarySearchTree & operator= ( const BinarySearchTree & rhs ); private: BinaryNode *root; const Comparable ITEM_NOT_FOUND;

29. 29 int main( ) { const int ITEM_NOT_FOUND = -9999; BinarySearchTree t( ITEM_NOT_FOUND ); int NUMS = 4000; const int GAP = 37; int i; for( i = GAP; i != 0; i = ( i + GAP ) % NUMS ) t.insert( i ); for( i = 1; i < NUMS; i+= 2 ) t.remove( i ); if( NUMS < 40 ) t.printTree( ); if( t.findMin( ) != 2 || t.findMax( ) != NUMS - 2 ) cout << "FindMin or FindMax error!" << endl; for( i = 2; i < NUMS; i+=2 ) if( t.find( i ) != i ) cout << "Find error1!" << endl; for( i = 1; i < NUMS; i+=2 ) { if( t.find( i ) != ITEM_NOT_FOUND ) cout << "Find error2!"<< endl;} BinarySearchTree t2( ITEM_NOT_FOUND ); t2 = t; for( i = 2; i < NUMS; i+=2 ) if( t2.find( i ) != i ) cout << "Find error1!" << endl; return 0; }

30. 30 template class BinarySearchTree { public: explicit BinarySearchTree( const Comparable & notFound); BinarySearchTree( const BinarySearchTree & rhs ); ~BinarySearchTree( ); const Comparable & findMin( ) const; const Comparable & findMax( ) const; const Comparable & find (const Comparable & x) const; bool isEmpty( ) const; void printTree( ) const; void makeEmpty( ); void insert( const Comparable & x ); void remove( const Comparable & x ); const BinarySearchTree & operator= ( const BinarySearchTree & rhs ); private: BinaryNode *root; const Comparable ITEM_NOT_FOUND;

31. 31 private: const Comparable & elementAt ( BinaryNode *t) const; void insert ( const Comparable & x, BinaryNode * & t ) const; void remove ( const Comparable & x, BinaryNode * & t ) const; BinaryNode * findMin ( BinaryNode *t ) const; BinaryNode * findMax ( BinaryNode *t ) const; BinaryNode * find ( const Comparable & x, BinaryNode *t ) const; void makeEmpty ( BinaryNode * & t ) const; void printTree ( BinaryNode *t ) const; BinaryNode * clone ( BinaryNode ) *t ) const; };

32. 32 find /* Find item x in the tree - Return the matching item or ITEM_NOT_FOUND if not found. */ template const Comparable & BinarySearchTree :: find( const Comparable & x ) const { return elementAt( find( x, root ) ); } template const Comparable & BinarySearchTree :: elementAt( BinaryNode *t ) const { if( t == NULL ) return ITEM_NOT_FOUND; else return t->element; }

33. 33 find

34. 34 find /* Find item x in the tree - Return the node containinig the matching item or NULL if not found. */ template BinaryNode * BinarySearchTree :: find ( const Comparable & x, BinaryNode *t ) const; { if (t==NULL) return NULL; else if (x element) return find( x, t->left ); else if (t->element < x) return find( x, t->right ); else retun t; }

35. 35 Recursive implementation of findMin /* Internal method to find the smallest item in a subtree t. */ template BinaryNode * BinarySearchTree ::findMin( BinaryNode *t ) const { if( t == NULL ) return NULL; if( t->left == NULL ) return t; return findMin( t->left ); }

36. 36 Non-recursive implementation of findMax template BinaryNode * BinarySearchTree ::findMax( BinaryNode *t ) const { if( t != NULL ) while( t->right != NULL ) t = t->right; return t; }

37. 37 insert into a binary search tree

38. 38 insert into a binary search tree template void BinarySearchTree :: insert( const Comparable & x, BinaryNode * & t ) const { if( t == NULL ) t = new BinaryNode ( x, NULL, NULL ); else if( x element ) insert( x, t->left ); else if( t->element < x ) insert( x, t->right ); else ; // Duplicate; do nothing }

39. 39 Remove

40. 40 Remove template void BinarySearchTree :: remove( const Comparable & x, BinaryNode * & t ) const { if( t == NULL ) return; if( x element ) remove( x, t->left ); else if( t->element right ); else if( t->left != NULL && t->right != NULL ) { t->element = findMin( t->right )->element; remove( t->element, t->right ); } else { BinaryNode *oldNode = t; t = ( t->left != NULL ) ? t->left : t->right; delete oldNode; }

41. 41 Destructor and recursive makeEmpty /* Destructor for the tree. */ template BinarySearchTree ::~BinarySearchTree( ){ makeEmpty( ); } template void BinarySearchTree :: makeEmpty( BinaryNode * & t ) const { if( t != NULL ) { makeEmpty( t->left ); makeEmpty( t->right ); delete t; } t = NULL; }

42. 42 Operator = template const BinarySearchTree & BinarySearchTree :: operator=( const BinarySearchTree & rhs ) { if( this != &rhs ){ makeEmpty( ); root = clone( rhs.root ); } return *this; }

43. 43 Recursive clone member function template BinaryNode * BinarySearchTree ::clone( BinaryNode * t ) const { if( t == NULL ) return NULL; else return new BinaryNode ( t->element, clone( t->left ), clone( t->right ) ); }