Tree

1. Tree

2. Outline • Tree Data Structure • Examples • Terminology & Definition • Binary Tree • Tree Traversal • Tree Iterators

3. Trees: Some Examples • A tree represents a hierarchy, for example: • Organizational structure of a company

4. Trees: Some Examples • Table of contents of a book

5. Trees: Some Examples • File system (Unix, Windows, Internet)

6. Trees: Terminology • Ais the root node • Bis the parentof D and E • Cis the siblingof B • Dand E are the children of B • D, E, F, G, I are external nodes, or leaves • A, B, C, Hare internal nodes • The depth, level, orpath length of Eis 2 • The heightof the tree is 3 • The degreeof node Bis 2 Property: |edges| = |nodes| - 1

7. Trees: Viewed Recursively A sub-tree is also a tree!!

8. Binary Trees • Binary tree: tree with all internal nodes of degree 2 • Recursive View: a binary tree is either • empty • an internal node (the root) and two binary trees(left subtree and right subtree)

9. Binary Trees: An Example • Arithmetic expression

10. Binary Trees: Properties • If we restrict that each parent can have two and only two children, then: • |external nodes| = |internal nodes| + 1 • |nodes at level i|  2i • |external nodes|  2 (height) • height  log2 |external nodes| • height  log2 |nodes| - 1 • height  |internal nodes| = (|nodes| - 1)/2

11. Traversing Trees: Preorder • Preorder Traversal Algorithm preOrder(v) “visit” node v for each child wof vdo recursively perform preOrder(w) • Example: reading a document from beginning to end

12. Traversing Trees: Postorder • Postorder Traversal AlgorithmpostOrder(v) for each child wof vdo recursively perform postOrder(w) “visit” node v • Example: du(disk usage) command in Unix

13. Traversing Trees • Algorithm evaluateExpression(v) if v is an external node return the variable stored at v else let o be the operator stored at v x evaluateExpression(leftChild(v)) y evaluateExpression(rightChild(v)) return x o y

14. Traversing Trees: Inorder • Inorder traversal of a binary tree Algorithm inOrder(v) recursively perform inOrder(leftChild(v)) “visit” node v recursively perform inOrder(rightChild(v)) • Example: printing an arithmetic expression • print “(“ before traversing the left subtree • print “)” after traversing the right subtree

15. The BinaryNode in Java • A tree is a collection of nodes: class BinaryNode <T> { T element; /* Item stored in node */ BinaryNode<T> left; BinaryNode<T> right; } • The tree stores a reference to the root node, which is the starting point. public class BinaryTree <T> { private BinaryNode<T> root; // Root node public BinaryTree() // Default constructor { root = null; } }

16. HR+1 HL+1 HR HL Think Recursively • Computing the height of a tree is complex without recursion. • The height of a tree is one more than the maximum of the heights of the subtrees. • HT = max (HL+1, HR+1)

17. Routine to Compute Height • Handle base case (empty tree) • Use previous observation for recursive case. public static int height (BinaryNode t) { if (t == null) return 0; else return 1 + max(height (t.left), height (t.right)); }

18. Running Time • This strategy is a postorder traversal: information for a tree node is computed after the information for its children is computed. • The running time of tree traversal is N times the cost of processing each node. • Thus, the running time is linear because we do constant work for each node in the tree.

19. Print Preorder class BinaryNode { void printPreOrder( ) { System.out.println( element ); // Node if( left != null ) left.printPreOrder( ); // Left if( right != null ) right.printPreOrder( ); // Right } } class BinaryTree { public void printPreOrder( ) { if( root != null ) root.printPreOrder( ); } }

20. Print Postorder class BinaryNode { void printPostOrder( ) { if( left != null ) left.printPostOrder( ); // Left if( right != null ) right.printPostOrder( ); // Right System.out.println( element ); // Node } } class BinaryTree { public void printPostOrder( ) { if( root != null ) root.printPostOrder( ); } }

21. Print Inorder class BinaryNode { void printInOrder( ) { if( left != null ) left.printInOrder( ); // Left System.out.println( element ); // Node if( right != null ) right.printInOrder( ); // Right } } class BinaryTree { public void printInOrder( ) { if( root != null ) root.printInOrder( ); } }

22. Traversing Tree Pre-Order Post-Order InOrder

23. Exercise • A tree contains Integer objects. • Find the maximum value • Find the total value

24. Tree Iterator Class • Can we implement traversal non-recursively? • Recursion is implemented by using a stack. • We can traverse non-recursively by maintaining the stack ourselves (emulate stack of activation records). • Is a non-recursive approach faster than recursive approach? • Possibly:we can place only the essentials, while the compiler places an entire activation record.

25. Postorder Traversal using Stack • Use stack to store the current state (nodes we have traversed but not yet completed) • Similar to PC (program counter) in the activation record • We “pop” each node three times, when: 0. about to make a recursive call to left subtree 1. about to make a recursive call to right subtree 2. about to process the current node itself

26. Postorder Algorithm • init: push root onto stack with state 0 • advance: • while (true) • node X = pop from the stack • switch (state X): • case 0: • push node X with state 1 • push left child of X (if it exists) with state 0 • case 1: • push node X with state 2 • push right child of X (if it exists) with state 0 • case 2: • visit/process node X

27. Postorder Traversal: Stack States

28. Exercise • Create a non-recursive algorithm for inorder traversal using stack. • Create a non-recursive algorithm for preordertraversal using stack.

29. Inorder Traversal using Stack • What are the states for inorder traversal? 0. about to make a recursive call to left subtree 1. about to process the current node 2. about to make a recursive call to right subtree

30. Inorder Algorithm • init: push root onto the stack with state 0 • advance: • while (true) • node X = pop from the stack • switch (state X): • case 0: • push node X with state 1 • push left child of X (if it exists) with state 0 • case 1: • push node X with state 2 • visit/process node X • case 2: • push right child of X (if it exists) with state 0

31. Inorder Algorithm (improved) • init: push root onto the stack with state 0 • advance (optimize): • while (true) • node X = pop from the stack • switch (state X): • case 0: • push node X with state 1 • push left child of X (if it exists) with state 0 • case 1: • visit/process node X • push right child of X (if it exists) with state 0

32. Preorder Traversal using Stack • What are the states for preorder traversal? 0. about to process the current node 1. about to make a recursive call to left subtree 2. about to make a recursive call to right subtree

33. Preorder Algorithm • init: push root onto the stack with state 0 • advance: • while (true) • node X = pop from the stack • switch (state X): • case 0: • push node X with state 1 • visit/process the node X • case 1: • push node X with state 2 • push left child of X (if it exists) with state 0 • case 2: • push right child of X (if it exists) with state 0

34. Preorder Algorithm (improved) • init: push root onto the stack • advance (optimized): • while (true) • node X = pop from the stack • visit/process node X • push right child of X (if it exists) • push left child of X (if it exists)

35. Euler Tour Traversal • Generic traversal of a binary tree • The preorder, inorder, and postorder traversals are special cases of the Euler tour traversal • “walk around” the tree and visit each node three times: • on the left • from below • on the right

36. Level-Order Traversal • Visit root followed by its children from left to right and followed by their children. So we go down the tree level by level. • Sequence: A - B - C - D - E - F - G - H - I

37. Level-Order Traversal: Idea • Using a queue instead of a stack • Algorithm (similar to pre-order) • init: enqueue the root into the queue • while(true): • node X = dequeue from the queue • visit/process node X; • enqueue left child of node X (if it exists); • enqueue right child of node X (if it exists);

38. Binary Tree: Properties • Maximum number of nodes in a binary tree of height k is 2k+1 -1. • A full binary tree with height k is a binary tree which has 2k+1 - 1 nodes. • A complete binary tree with height k is a binary tree which has maximum number of nodes possible in levels 0 through k -1, and in (k -1)’th level all nodes with children are selected from left to right. • Complete binary tree with n nodes can be shown by using an array, then for any node with index i, we have: • Parent (i) is at i/2 if i 1; for i =1, we have no parent. • Left-child (i ) is at 2i if 2i  n. (else no left-child) • Right-child (i ) is at 2i+1 if 2i +1  n (else no right-child)

39. Summary • Tree, Binary Tree • In order to process the elements of a tree, we consider accessing the elements in certain order • Tree traversal is a tree operation that involves "visiting” (or" processing") all the nodes in a tree. • Depth First Search (DFS): • Pre-order: Visit node first, pre-order all its subtrees from leftmost to rightmost. • Inorder: Inorder the node in left subtree and then visit the root following by inorder traversal of all its right subtrees. • Post-order: Post-order the node in left subtree and then post-order the right subtrees followed by visit to the node. • Breadth First Search (BFS): • Level-order: Visit root followed by its children from left to right and followed by their children. So we go down the tree level by level.