Homework Assignment #3

Transcription

1 CISC 2200 Data Structure Spring, 2016 Homework Assignment #3 1 Which of these formulas gives the maximum total number of nodes in a binary tree that has N levels? (Remember that the root is Level 0.) Explain your answer. (a) N 2 1 (b) 2 N (c) 2 N 1 (d) 2 N+1 2 Which of these formulas gives the maximum number of nodes in the N-the level of a binary tree? (a) N 2 (b) 2 N (c) 2 N 1 (d) 2 N+1 3 How many ancestors does a node in the N-th level of a binary tree have?

2 4 Answer the following questions about binary search tree, assuming elements are inserted into the tree in the following order. The implementation of PutItem and DeleteItem are given below: 50, 72, 96, 94, 107, 26, 12, 11, 9, 2, 10, 25, 51, 16, 17, 95 void Retrieve(TreeNode* tree, ItemType& item, bool& found); ItemType TreeType::GetItem(ItemType item, bool& found) // Calls recursive function Retrieve to search the tree for item. Retrieve(root, item, found); return item; void Retrieve(TreeNode* tree, ItemType& item, bool& found) // Recursively searches tree for item. // Post: If there is an element someitem whose key matches item s, // found is true and item is set to a copy of someitem; // otherwise found is false and item is unchanged. if (tree == NULL) found = false; // item is not found. if (item < tree->info) Retrieve(tree->left, item, found); // Search left subtree. if (item > tree->info) Retrieve(tree->right, item, found);// Search right subtree. item = tree->info; // item is found. found = true; void Insert(TreeNode*& tree, ItemType item); void TreeType::PutItem(ItemType item) // Calls recursive function Insert to insert item into tree. Insert(root, item); void Insert(TreeNode*& tree, ItemType item) // Inserts item into tree. // Post: item is in tree; search property is maintained. if (tree == NULL) // Insertion place found. 2

3 tree = new TreeNode; tree->right = NULL; tree->left = NULL; tree->info = item; if (item < tree->info) Insert(tree->left, item); Insert(tree->right, item); void DeleteNode(TreeNode*& tree); void Delete(TreeNode*& tree, ItemType item); // Insert in left subtree. // Insert in right subtree. void TreeType::DeleteItem(ItemType item) // Calls recursive function Delete to delete item from tree. Delete(root, item); void Delete(TreeNode*& tree, ItemType item) // Deletes item from tree. // Post: item is not in tree. if (item < tree->info) Delete(tree->left, item); // Look in left subtree. if (item > tree->info) Delete(tree->right, item); // Look in right subtree. DeleteNode(tree); // Node found; call DeleteNode. void GetPredecessor(TreeNode* tree, ItemType& data); void DeleteNode(TreeNode*& tree) // Deletes the node pointed to by tree. // Post: The user s data in the node pointed to by tree is no // longer in the tree. If tree is a leaf node or has only // non-null child pointer the node pointed to by tree is // deleted; otherwise, the user s data is replaced by its // logical predecessor and the predecessor s node is deleted. ItemType data; TreeNode* tempptr; tempptr = tree; if (tree->left == NULL) tree = tree->right; delete tempptr; 3

4 if (tree->right == NULL) tree = tree->left; delete tempptr; GetPredecessor(tree->left, data); tree->info = data; Delete(tree->left, data); // Delete predecessor node. void GetPredecessor(TreeNode* tree, ItemType& data) // Sets data to the info member of the right-most node in tree. while (tree->right!= NULL) tree = tree->right; data = tree->info; (a) Draw the binary tree after the above numbers are inserted in the given order. (b) Show the order in which the nodes in the tree are processed by: an in-order traversal of the tree. a post-order traversal of the tree 4

5 a preorder traversal of the tree (c) What s the minimum height of the binary search tree storing these numbers (not necessarily inserted in the given order)? Draw a tree with the minimum height below: (d) Trace the DeleteItem function when called to delete numbers 95, 25, and then If you want to traverse a tree, writing all the elements to a file, and later (the next time you run the program) rebuild the tree by reading and inserting, would an inoder traversal be appropriate? Why or why not? 5

6 6 Write a member function IsBST to the class TreeType that determines whether a binary tree is a binary search tree. truct TreeNode ItemType info; TreeNode* left; TreeNode* right; ; class TreeType public:... private: TreeNode* root;... ; (a) Write the recursive implementation of this function. (b) Could you also write an iterative implementation of this function? 7 A binary tree is stored in an array called TreeNodes, which is indexed from 0 to 99, as described in the chapter. The tree cotnains 85 elements. Mark each of the following statements as True or False, and correct any false statements. (a) treenodes[42] is a leaf node. (b) treenodes[41] has only one child. 6

7 (c) The right child of treenodes[12] is treenodes[25]. (d) The subtree rooted at treenodes[7] is a full binary tree with four levels. (e) The tree has seven levels that are full, and one additional level that contains some elements. 7