Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 13.2, Problem 2E
Program Plan Intro
To argue that every n node binary search tree, there are exactly
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Draw the binary search tree that would result if the given elements were added to an empty binary search tree in the given order. Use paint to draw.
a. Lisa, Bart, marge, maggie, flanders, smithers, miljouse
b. 12,34,1,5,-5,6,19,45,2,-7,47
1. Find the Big-O time complexity for the following:-Inserting an element in an (unbalanced) binary search tree? And what is the best case?-Best case for the height of an (unbalanced) binary search tree? And what is the average case big-O given the avg height if elements are inserted in random order?2. In a binary search tree of N nodes, how many subtrees are there?
Let T be a binary search tree whose keys are all distinct. Suppose T is created by inserting each key into the growing tree. Let x be a leaf node and y be its parent node. Show that if x is the right child of y, then y.key is the largest key in T smaller than x.key. (Hint: Show that the assumption that there is another node z whose key satisfies y.key < z.key < x.key will lead to a contradiction.)
Chapter 13 Solutions
Introduction to Algorithms
Ch. 13.1 - Prob. 1ECh. 13.1 - Prob. 2ECh. 13.1 - Prob. 3ECh. 13.1 - Prob. 4ECh. 13.1 - Prob. 5ECh. 13.1 - Prob. 6ECh. 13.1 - Prob. 7ECh. 13.2 - Prob. 1ECh. 13.2 - Prob. 2ECh. 13.2 - Prob. 3E
Ch. 13.2 - Prob. 4ECh. 13.2 - Prob. 5ECh. 13.3 - Prob. 1ECh. 13.3 - Prob. 2ECh. 13.3 - Prob. 3ECh. 13.3 - Prob. 4ECh. 13.3 - Prob. 5ECh. 13.3 - Prob. 6ECh. 13.4 - Prob. 1ECh. 13.4 - Prob. 2ECh. 13.4 - Prob. 3ECh. 13.4 - Prob. 4ECh. 13.4 - Prob. 5ECh. 13.4 - Prob. 6ECh. 13.4 - Prob. 7ECh. 13 - Prob. 1PCh. 13 - Prob. 2PCh. 13 - Prob. 3PCh. 13 - Prob. 4P
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- Is the following statement true? Justify your answer. Let B be a binary tree. If for each of its vertices 'v' a data item is inserted into 'v' bigger, than the data item inserted in the left son of the vertex 'v', and at the same time smaller than data item inserted in the right son of the vertex 'v', then B is a search tree.arrow_forwardIllustrate that via AVL single rotation, any binary search tree T1 can betransformed into another search tree T2 (with the same items) Give an algorithm to perform this transformation using O(N log N) rotation on averagearrow_forwardA) Suppose the numbers 7 , 5 , 1 , 8 , 3 , 6 , 0 , 9 , 4 , 2 are inserted in that order into an initially empty binary search tree. The binary search tree uses the usual ordering on natural numbers.ShowA) In order B) pre order C) post order B)From the given tree state predict1. Degree of tree2. Degree of node G3. Level of node H4. Prove that formula to find maximun number of node is not applicable here , along with reason5. Post order, pre order and in order traversingarrow_forward
- (4) Create a binary search tree as discussed in class, using the given numbers in the order they’re presented. State if the resulting tree is has the attributes of being FULL, BALANCED, or COMPLETE. 37, 20, 18, 56, 40, 42, 12, 5, 6, 77, 21, 54arrow_forwardUse the procedure TREE-SUCCESSOR and TREE-MINIMUM to write a function of x, x is a node in a binary search tree, to produce the output that INORDERTREE-WALK function would produce. Determine the upper bound running time complexity of F(x) and prove its correctness.arrow_forwardDraw a binary search tree and AVL tree from the following traversals: {15, 5, 20, 70, 3, 10, 60, 90, 16} Convert Binary Search Tree into Binary Tree such that sum of all greater keys is added to every key.arrow_forward
- The time complexity of traversing a binary search tree that contains 'n' elements is O (n) in the worst case.The time complexity of traversing a red-black tree that contains 'n' elements is O (log2 n) in the worst case.Traversing a binary search tree, which contains integers, according to the "inorder" principle always gives us integers in sorted ascending order.If we insert a sequence of integers in an empty BST, it will always be at least as high as the tree we get if we insert the same sequence of integers in an empty red-black tree.If we insert a sequence of integers in an empty BST, it will always be twice as high as the tree we get if we insert the same sequence of integers in an empty red-black tree.Group of answer options Only statements A, B, C and D are correct. Only statements A, B and C are correct. All statements are correct. Only statements A, C, D and E are correct. Only statements A, C and D are correct.arrow_forwardConsider a binary search tree (BST) whose elements are the integer values: Starting with an empty BST, show the effect of successively adding the following elements: 15, 78, 10, 5, 12, 82, 60, 68, 11 and 65. Show the effect of successively deleting 10, 78, and 15 from the resulting BST. Traverse the original BST in pre-order and post-order.arrow_forwardConsider the Binary Search Tree method DepthEqual(T) that outputs all node values in a binary search tree T such that the value stored in the node is equal to the depth of the node in the tree. The values can be any integer, including negative numbers. Show pseudocode for an efficient implementation of DepthEqual(T). Hint: For a node n at depth d, whose value is v. if d >= v. then there is no need to search n's left subtreearrow_forward
- Let T be an arbitrary splay tree storing n elements A1, A2, . An, where A1 ≤ A2 ≤ . . . ≤ An. We perform n search operations in T, and the ith search operation looks for element Ai. That is, we search for items A1, A2, . . . , An one by one. What will T look like after all these n operations are performed? For example, what will the shape of the tree be like? Which node stores A1, which node stores A2, etc.? Prove the answer you gave for formally. Your proof should work no matter what the shape of T was like before these operations.arrow_forwardWe say that a binary search tree T1 can be right-converted to binary search tree T2 if it is possible to obtain T2 from T1 via a series of right rotations. Please draw diagram and Give an example of two trees T1 and T2 such that T1 cannot be right-converted to T2. Explain your answer.arrow_forwardA binary search tree may be balanced or unbalanced based on the arrangement of thenodes of the tree. With your knowledge in Binary search tree:i. Explain the best and worst-case scenarios of the time and space complexity of both types ofbinary trees abovearrow_forward
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