1. CS 1031 Tree Traversal Techniques; Heaps Tree Traversal Concept Tree Traversal Techniques: Preorder, Inorder, Postorder Full Trees Almost Complete Trees Heaps

2. CS 1032 Binary-Tree Related Definitions The children of any node in a binary tree are ordered into a left child and a right child A node can have a left and a right child, a left child only, a right child only, or no children The tree made up of a left child (of a node x) and all its descendents is called the left subtree of x Right subtrees are defined similarly 10 1 3 11 98 46 5 7 12

3. CS 1033 A Binary-tree Node Class class TreeNode { public: typedef int datatype; TreeNode(datatype x=0, TreeNode *left=NULL, TreeNode *right=NULL){ data=x; this->left=left; this->right=right; }; datatype getData( ) {return data;}; TreeNode *getLeft( ){return left;}; TreeNode *getRight( ){return right;}; void setData(datatype x) {data=x;}; void setLeft(TreeNode *ptr) {left=ptr;}; void setRight(TreeNode *ptr) {right=ptr;}; private: datatype data; // different data type for other apps TreeNode *left; // the pointer to left child TreeNode *right; // the pointer to right child };

4. CS 1034 Binary Tree Class class Tree { public: typedef int datatype; Tree(TreeNode *rootPtr=NULL){this->rootPtr=rootPtr;}; TreeNode *search(datatype x); bool insert(datatype x); TreeNode * remove(datatype x); TreeNode *getRoot(){return rootPtr;}; Tree *getLeftSubtree(); Tree *getRightSubtree(); bool isEmpty(){return rootPtr == NULL;}; private: TreeNode *rootPtr; };

5. CS 1035 Binary Tree Traversal Traversal is the process of visiting every node once Visiting a node entails doing some processing at that node, but when describing a traversal strategy, we need not concern ourselves with what that processing is

6. CS 1036 Binary Tree Traversal Techniques Three recursive techniques for binary tree traversal In each technique, the left subtree is traversed recursively, the right subtree is traversed recursively, and the root is visited What distinguishes the techniques from one another is the order of those 3 tasks

7. CS 1037 Preoder, Inorder, Postorder In Preorder, the root is visited before (pre) the subtrees traversals In Inorder, the root is visited in-between left and right subtree traversal In Preorder, the root is visited after (pre) the subtrees traversals Preorder Traversal: 1.Visit the root 2.Traverse left subtree 3.Traverse right subtree Inorder Traversal: 1.Traverse left subtree 2.Visit the root 3.Traverse right subtree Postorder Traversal: 1.Traverse left subtree 2.Traverse right subtree 3.Visit the root

8. CS 1038 Illustrations for Traversals Assume: visiting a node is printing its label Preorder: 1 3 5 4 6 7 8 9 10 11 12 Inorder: 4 5 6 3 1 8 7 9 11 10 12 Postorder: 4 6 5 3 8 11 12 10 9 7 1 1 3 11 98 46 5 7 12 10

9. CS 1039 Illustrations for Traversals (Contd.) Assume: visiting a node is printing its data Preorder: 15 8 2 6 3 7 11 10 12 14 20 27 22 30 Inorder: 2 3 6 7 8 10 11 12 14 15 20 22 27 30 Postorder: 3 7 6 2 10 14 12 11 8 22 30 27 20 15 6 15 8 2 37 11 10 14 12 20 27 22 30

10. CS 10310 Code for the Traversal Techniques The code for visit is up to you to provide, depending on the application A typical example for visit(…) is to print out the data part of its input node void inOrder(Tree *tree){ if (tree->isEmpty( )) return; inOrder(tree->getLeftSubtree( )); visit(tree->getRoot( )); inOrder(tree->getRightSubtree( )); } void preOrder(Tree *tree){ if (tree->isEmpty( )) return; visit(tree->getRoot( )); preOrder(tree->getLeftSubtree()); preOrder(tree->getRightSubtree()); } void postOrder(Tree *tree){ if (tree->isEmpty( )) return; postOrder(tree->getLeftSubtree( )); postOrder(tree->getRightSubtree( )); visit(tree->getRoot( )); }

11. CS 10311 Application of Traversal Sorting a BST Observe the output of the inorder traversal of the BST example two slides earlier It is sorted This is no coincidence As a general rule, if you output the keys (data) of the nodes of a BST using inorder traversal, the data comes out sorted in increasing order

12. CS 10312 Other Kinds of Binary Trees (Full Binary Trees) Full Binary Tree: A full binary tree is a binary tree where all the leaves are on the same level and every non-leaf has two children The first four full binary trees are:

13. CS 10313 Examples of Non-Full Binary Trees These trees are NOT full binary trees: (do you know why?)

14. CS 10314 Canonical Labeling of Full Binary Trees Label the nodes from 1 to n from the top to the bottom, left to right 1 1 2 3 1 2 3 45 6 7 1 23 4 56 7 8910 11 12 131415 Relationships between labels of children and parent: 2i 2i+1 i

15. CS 10315 Other Kinds of Binary Trees (Almost Complete Binary trees) Almost Complete Binary Tree: An almost complete binary tree of n nodes, for any arbitrary nonnegative integer n, is the binary tree made up of the first n nodes of a canonically labeled full binary 1 1 2 1 23 45 6 7 1 2 1 23 45 6 1 23 4 1 23 45

16. CS 10316 Depth/Height of Full Trees and Almost Complete Trees The height (or depth ) h of such trees is O(log n) Proof: In the case of full trees, –The number of nodes n is: n=1+2+2 2 +2 3 +…+2 h =2 h+1 -1 –Therefore, 2 h+1 = n+1, and thus, h=log(n+1)-1 –Hence, h=O(log n) For almost complete trees, the proof is left as an exercise.

17. CS 10317 Canonical Labeling of Almost Complete Binary Trees Same labeling inherited from full binary trees Same relationship holding between the labels of children and parents: Relationships between labels of children and parent: 2i 2i+1 i

18. CS 10318 Array Representation of Full Trees and Almost Complete Trees A canonically label-able tree, like full binary trees and almost complete binary trees, can be represented by an array A of the same length as the number of nodes A[k] is identified with node of label k That is, A[k] holds the data of node k Advantage: –no need to store left and right pointers in the nodes  save memory –Direct access to nodes: to get to node k, access A[k]

19. CS 10319 Illustration of Array Representation Notice: Left child of A[5] (of data 11) is A[2*5]=A[10] (of data 18), and its right child is A[2*5+1]=A[11] (of data 12). Parent of A[4] is A[4/2]=A[2], and parent of A[5]=A[5/2]=A[2] 6 15 8 211 1812 20 27 13 30 15 820 2 11 30 27 13 610 12 1234567 891011

20. CS 10320 Adjustment of Indexes Notice that in the previous slides, the node labels start from 1, and so would the corresponding arrays But in C/C++, array indices start from 0 The best way to handle the mismatch is to adjust the canonical labeling of full and almost complete trees. Start the node labeling from 0 (rather than 1). The children of node k are now nodes (2k+1) and (2k+2), and the parent of node k is (k-1)/2, integer division.

21. CS 10321 Application of Almost Complete Binary Trees: Heaps A heap (or min-heap to be precise) is an almost complete binary tree where –Every node holds a data value (or key) –The key of every node is ≤ the keys of the children Note: A max-heap has the same definition except that the Key of every node is >= the keys of the children

22. CS 10322 Example of a Min-heap 16 5 8 1511 1812 20 27 33 30

23. CS 10323 Operations on Heaps Delete the minimum value and return it. This operation is called deleteMin. Insert a new data value Applications of Heaps: A heap implements a priority queue, which is a queue that orders entities not a on first-come first-serve basis, but on a priority basis: the item of highest priority is at the head, and the item of the lowest priority is at the tail Another application: sorting, which will be seen later

24. CS 10324 DeleteMin in Min-heaps The minimum value in a min-heap is at the root! To delete the min, you can’t just remove the data value of the root, because every node must hold a key Instead, take the last node from the heap, move its key to the root, and delete that last node But now, the tree is no longer a heap (still almost complete, but the root key value may no longer be ≤ the keys of its children

25. CS 10325 Illustration of First Stage of deletemin 16 5 8 1511 1812 20 27 33 30 16 8 1511 18 12 20 27 33 30 16 8 1511 18 12 20 27 33 30 16 8 1511 18 12 20 27 33 30

26. CS 10326 Restore Heap To bring the structure back to its “heapness”, we restore the heap Swap the new root key with the smaller child. Now the potential bug is at the one level down. If it is not already ≤ the keys of its children, swap it with its smaller child Keep repeating the last step until the “bug” key becomes ≤ its children, or the it becomes a leaf

27. CS 10327 Illustration of Restore-Heap 16 8 1511 18 12 20 27 33 30 16 12 1511 18 8 20 27 33 30 16 11 1512 18 8 20 27 33 30 Now it is a correct heap

28. CS 10328 Time complexity of insert and deletmin Both operations takes time proportional to the height of the tree –When restoring the heap, the bug moves from level to level until, in the worst case, it becomes a leaf (in deletemin) or the root (in insert) –Each move to a new level takes constant time –Therefore, the time is proportional to the number of levels, which is the height of the tree. But the height is O(log n) Therefore, both insert and deletemin take O(log n) time, which is very fast.

29. CS 10329 Inserting into a minheap Suppose you want to insert a new value x into the heap Create a new node at the “end” of the heap (or put x at the end of the array) If x is >= its parent, done Otherwise, we have to restore the heap: –Repeatedly swap x with its parent until either x reaches the root of x becomes >= its parent

30. CS 10330 Illustration of Insertion Into the Heap In class

31. CS 10331 The Min-heap Class in C++ class Minheap{ //the heap is implemented with a dynamic array public: typedef int datatype; Minheap(int cap = 10){capacity=cap; length=0; ptr = new datatype[cap];}; datatype deleteMin( ); void insert(datatype x); bool isEmpty( ) {return length==0;}; int size( ) {return length;}; private: datatype *ptr; // points to the array int capacity; int length; void doubleCapacity(); //doubles the capacity when needed };

32. CS 10332 Code for deletemin Minheap::datatype Minheap::deleteMin( ){ assert(length>0); datatype returnValue = ptr[0]; length--; ptr[0]=ptr[length]; // move last value to root element int i=0; while ((2*i+1 ptr[2*i+1]) || (2*i+2 ptr[2*i+1] || ptr[i]>ptr[2*i+2]))){ // “bug” still > at least one child if (ptr[2*i+1] <= ptr[2*i+2]){ // left child is the smaller child datatype tmp= ptr[i]; ptr[i]=ptr[2*i+1]; ptr[2*i+1]=tmp; //swap i=2*i+1; } else{ // right child if the smaller child. Swap bug with right child. datatype tmp= ptr[i]; ptr[i]=ptr[2*i+2]; ptr[2*i+2]=tmp; // swap i=2*i+2; } } return returnValue; };

33. CS 10333 Code for Insert void Minheap::insert(datatype x){ if (length==capacity) doubleCapacity(); ptr[length]=x; int i=length; length++; while (i>0 && ptr[i] < ptr[i/2]){ datatype tmp= ptr[i]; ptr[i]=ptr[(i-1)/2]; ptr[(i-1)/2]=tmp; i=(i-1)/2; } };

34. CS 10334 Code for doubleCapacity void Minheap::doubleCapacity(){ capacity = 2*capacity; datatype *newptr = new datatype[capacity]; for (int i=0;i<length;i++) newptr[i]=ptr[i]; delete [] ptr; ptr = newptr; };