1. Lecture12: Tree II Bohyung Han CSE, POSTECH bhhan@postech.ac.kr CSED233: Data Structures (2014F)

2. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Tree Traversal Process of visiting nodes in a tree systematically  Some algorithms need to visit all nodes in a tree.  Example: printing, counting nodes, etc. Implementation  Can be done easily by recursion  Order of visits does matter. 2 Computers”R”Us SalesR&DManufacturing LaptopsDesktops US International EuropeAsiaCanada

3. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Preorder Traversal A preorder traversal of the subtree rooted at node n:  Visit node n (process the node's data).  Recursively perform a preorder traversal of the left child.  Recursively perform a preorder traversal of the right child. A preorder traversal starts at the root. 3 public void preOrderTraversal(Node n) { if (n == null) return; System.out.print(n.value+" "); preOrderTraversal(n.left); preOrderTraversal(n.right); }

4. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Preorder Traversal 4 1 2 34 56 7 8 9 public void preOrderTraversal(Node n) { if (n == null) return; System.out.print(n.value+" "); preOrderTraversal(n.left); preOrderTraversal(n.right); }

5. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Inorder Traversal A preorder traversal of the subtree rooted at node n:  Recursively perform an inorder traversal of the left child.  Visit node n (process the node's data).  Recursively perform an inorder traversal of the right child. An iorder traversal starts at the root. 5 public void inOrderTraversal(Node n) { if (n == null) return; inOrderTraversal(n.left); System.out.print(n.value+" "); inOrderTraversal(n.right); } Defined only in binary tree.

6. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Inorder Traversal 6 6 2 14 35 7 9 8 public void inOrderTraversal(Node n) { if (n == null) return; inOrderTraversal(n.left); System.out.print(n.value+" "); inOrderTraversal(n.right); }

7. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Print Arithmetic Expressions Specialization of an inorder traversal  print operand or operator when visiting node  print “(“ before traversing left subtree  print “)“ after traversing right subtree 7 Algorithm printExpression(v) if hasLeft (v) print(“(’’) printExpression (left(v)) print(v.element ()) if hasRight (v) printExpression (right(v)) print (“)’’) +  - 2 a1 3b 3 1 2 5 6 79 8 4 ( ( ( ) ) () )

8. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Postorder Traversal A preorder traversal of the subtree rooted at node n:  Recursively perform a postorder traversal of the left child.  Recursively perform a postorder traversal of the right child.  Visit node n (process the node's data). A postorder traversal starts at the root. 8 public void postOrderTraversal(Node n) { if (n == null) return; postOrderTraversal(n.left); postOrderTraversal(n.right); System.out.print(n.value+" "); }

9. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Postorder Traversal 9 9 5 14 23 8 7 6 public void postOrderTraversal(Node n) { if (n == null) return; postOrderTraversal(n.left); postOrderTraversal(n.right); System.out.print(n.value+" "); }

10. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Evaluate Arithmetic Expressions Specialization of a postorder traversal  Recursive method returning the value of a subtree  When visiting an internal node, combine the values of the subtrees 10 Algorithm evalExpr(v) if isExternal (v) return v.element () else x  evalExpr(leftChild (v)) y  evalExpr(rightChild (v))   operator stored at v return x  y +  - 2 51 32 3 1 2 5 67 9 8 4 2 5 1 ‐ x 3 2 x +

11. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Binary Search Trees A binary search tree is a binary tree storing keys at its nodes and satisfying the following property:  Let u, v, and w be three nodes such that u is the left child v and w is the right child of v. Then, we have key(u)  key(v)  key(w).  The key value in v is larger than all keys in its left subtree and smaller than all keys in its right subtree. An inorder traversal of a binary search trees visits the keys in an increasing order. 11 6 92 418 v u w

12. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Search Searching a key  To search for a key k, we trace a downward path starting at the root  The next node visited depends on the comparison of k with the key of the current node  Search until we reach a leaf. Example: get(4)  Call TreeSearch(4, root) The algorithms for floorEntry and ceilingEntry are similar. 12 Algorithm TreeSearch(k, v) if T.isExternal (v) return v if k < key(v) return TreeSearch(k, T.left(v)) else if k = key(v) return v else return TreeSearch(k, T.right(v)) 6 9 2 4 1 8   

13. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Insertion Inserting a node  To perform operation insert(k), we search for key k using TreeSearch  Assume k is not already in the tree, and let w be the leaf reached by the search  We insert k at node w and expand w into an internal node Example: insert 5 13 6 9 2 4 18    w 6 92 418 5 w

14. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Insertion 14 public void insert(int i) { root = recursiveInsert(root, i); } private Node recursiveInsert(Node n, int i) { if (n == null) return new Node(i); if (i < n.value) { // insert in the left subtree n.left = recursiveInsert(n.left,i); return n; } else { // insert in the right subtree n.right = recursiveInsert(n.right,i); return n; }

15. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Deletion Deleting a node with a child  To perform operation remove(k), we search for key k  Assume key k is in the tree, and let v be the node storing k  Simply connect parent and child of v. Example: remove 4 15 6 9 2 4 18 5 v w 6 9 2 5 18  

16. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Deletion (cont.) Deleting a node with two children  We consider the case where the key k to be removed is stored at a node v whose children are both internal.  We find the internal node w that follows v in an inorder traversal.  we copy key(w) into node v.  we remove node w. Example: remove 3 16 3 1 8 6 9 5 v w z 2 5 1 8 6 9 v 2

17. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Implementation of Deletion 17 public void delete(int i) { root = recursiveRemove(root,i); } private Node recursiveRemove(Node n, int i) { if (n == null) // end of tree (node not found) return null; if (i < n.value) { // recurse left n.left = recursiveRemove(n.left,i); return n; } else if (i > n.value) { // recurse right n.right = recursiveRemove(n.right,i); return n; }

18. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Implementation of Deletion 18 else { // match if (n.left == null && n.right == null) { // no child return null; } else if (n.left != null && n.right == null) { // left child only return n.left; } else if (n.left == null && n.right != null) { // right child only return n.right; } else { // two children Node maxLeft = findMax(n.left); // find node to replace recursiveRemove(n.left, maxLeft.value); // remove the node maxLeft.left = n.left; // set children of replacement node maxLeft.right = n.right; return maxLeft; // return replacement node (adds it to tree) }

19. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Performance 19

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