Partha Sarathi Mandal

Transcription

1 MA 252: Data Structures and Algorithms Lecture 16 Partha Sarathi Mandal Dept. of Mathematics, IIT Guwahati

2 Deletion in BST Three cases Case 1: If deleted node z has no children Its parent will be marked with that child as NIL Case 2: If z has one children Its parent and the child are directly linked (and z removed) Case 3: If z has two children We replace the value of z with either its in order successor (the left most child of the right subtree) or the in order predecessor (the right most child of the left subtree).

3 Remove From A Leaf Remove a leaf element. key = 7

4 Remove From A Leaf (contd.) Remove a leaf element. key = 35

5 Remove From A Degree 1 Node Remove from a degree 1 node. key = 40

6 Remove From A Degree 1 Node (contd.) Remove from a degree 1 node. key = 15

7 Remove From A Degree 2 Node Remove from a degree 2 node. key = 10

8 Replace with largest key in left subtree (or smallest in right subtree). Remove From A Degree 2 Node

9 Replace with largest key in left subtree (or smallest in right subtree). Remove From A Degree 2 Node

10 Replace with largest key in left subtree (or smallest in right subtree). Remove From A Degree 2 Node

11 Largest key must be in a leaf or degree 1 node. Remove From A Degree 2 Node

12 Another Remove From A Degree 2 Node Remove from a degree 2 node. key = 20

13 Remove From A Degree 2 Node Replace with largest in left subtree.

14 Remove From A Degree 2 Node Replace with largest in left subtree.

15 Remove From A Degree 2 Node Replace with largest in left subtree.

16 Remove From A Degree 2 Node Complexity is O(height).

17 Deletion in BST TREE-DELETE(T, z) if left[z] = NIL or right[z] = NIL then //Case 1 or Case 2 y z else y TREE-SUCCESSOR(z) //Case 3 if left[y] NIL then x left[y] 10 else x right[y] if x NIL then p[x] p[y] if p[y] = NIL then root[t] x else if y == left[p[y]] then 6 15 left[p[y]] x else right[p[y]] x if y z then 2 8 key[z] key[y] 7 return y 18 p[y] 20 x z,y 40

18 Deletion in BST TREE-DELETE(T, z) if left[z] = NIL or right[z] = NIL then //Case 1 or Case 2 y z else y TREE-SUCCESSOR(z) //Case 3 if left[y] NIL then x left[y] 10 else x right[y] if x NIL then p[x] p[y] 6 15 if p[y] = NIL then root[t] x else if y == left[p[y]] then left[p[y]] x 2 8 else right[p[y]] x if y z then key[z] key[y] 7 return y 18 z p[y] 30 y x 28 40

19 Binary Tree Implementation Class Node { int data; // Could be int, a class, etc Node *left, *right, *parent; // null if empty } void insert ( int data ) { } void delete ( int data ) { } Node *find ( int data ) { }

20 Binary Tree Traversal Methods In a traversal of a binary tree, each element of the binary tree is visited exactly once. During the visit of an element, all action (make a clone, display, evaluate the operator, etc.) with respect to this element is taken.

21 Binary Tree Traversal Methods Pre-order In-order Post-order Level order

22 Preorder Traversal preorder(binarytreenode t){ if (t!= null){ visit(t); preorder(t.leftchild); preorder(t.rightchild); } }

23 Preorder Example (visit = print) a preorder(binarytreenode t){ if (t!= null){ visit(t); preorder(t.leftchild); preorder(t.rightchild); } } b a b c c

24 Preorder Example (visit = print) a b d e g h i f c j a b d g h e i c f j

25 Preorder Of Expression Tree / * e f a b c d / * + a b - c d + e f Gives prefix form of expression!

26 Inorder Traversal inorder(binarytreenode t){ if (t!= null){ inorder(t.leftchild); visit(t); inorder(t.rightchild); } }

27 Inorder Example (visit = print) inorder(binarytreenode t){ if (t!= null){ inorder(t.leftchild); visit(t); inorder(t.rightchild); } } b a b a c c

28 Inorder Example (visit = print) a b d e g h i f c j g d h b e i a f j c

29 Inorder By Projection (Squishing) a b d e g h i f c j g d h b e i a f j c

30 Inorder Of Expression Tree / * e f a b c d a + b * c - d / e + f Gives infix form of expression (sans parentheses)!

31 Postorder Traversal postorder(binarytreenode t){ if (t!= null){ postorder(t.leftchild); postorder(t.rightchild); visit(t); } }

32 Postorder Example (visit = print) postorder(binarytreenode t){ if (t!= null){ postorder(t.leftchild); postorder(t.rightchild); visit(t); } } b a b c a c

33 Postorder Example (visit = print) a b d e g h i f c j g h d i e b j f c a

34 Postorder Of Expression Tree / * e f a b c d a b + c d - * e f + / Gives postfix form of expression!

35 Traversal Applications a b d e g h i f c j Make a clone. Determine height. Determine number of nodes.

36 Level Order Let t be the tree root. (t!= null){ visit t and put its children on a FIFO queue; remove a node from the FIFO queue and call it t; // remove returns null when queue is empty }

37 Level-Order Example (visit = print) a b d e g h i f c j a b c d e f g h i j