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 14.2, Problem 3E
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If n1, n, ., nk is a sequence of nodes in the tree such that n; is the parent of
n41 for 1
2. We are given a complete binary tree with height h and n nodes. The link between a node and its left child is labeled as 0 and the link between a node and its right child is labeled as 1. A path from the root to each external node at the last level can be labeled by an h-tuple (X1, X2, ..., xh) of 1s and Os that lie on its links. See the following example:
0
0
1
1
0
0
1
0
(0,0,0)
(0,1,1)
Given a binary tree, write a function to check if it is a valid binary search tree (BST). Discuss the
approach you would take to validate the BST properties and handle edge cases.
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- Design a nonrecursive implementation of a binary tree that maintainsnode data in a Vector, data. In this implementation, element 0 of data references the root (if it exists). Every non-null element i of data finds its left andright children at locations 2i + 1 and 2(i + 1), respectively. (The inverse of theseindex relations suggests the parent of a nonroot node at i is found at locationb(i − 1)/2c.) Any element of data that is not used to represent a node shouldmaintain a null reference. don't copy answer from any ai tool otherwise you will get downvotearrow_forward1. A Preorder traversal of a binary tree (not necessarily a BST) produced A B D H I E J C F M and an Inorder traversal of the same tree produced H D I B J E A F M C. Upload a picture (it can be hand drawn) of the binary tree that will result to the previously mentioned traversals. 2. Explain how an internal node of a Binary Search Tree (BST) is deleted. Take note that after the deletion, the result is still a BST.arrow_forwardTl and T2 are two very huge binary trees, with Tl being much larger than T2. Develop an algorithm to detect whether T2 is a subtree of Tl.A tree T2 is a subtree of Tl if and only if there is a node n in Tl whose subtree is identical to T2.That is, if the tree were severed at node n, the two trees would be identical.arrow_forward
- The mapping approach that converts a full binary tree to a vector can be used to store generic trees, but in an inefficient manner. The method is to reserve enough space to keep the lowest, rightmost leaf and to keep null references in nodes that are not currently in use. What is the smallest possible Vector length required to store an n-element binary tree?arrow_forwardIn the BinaryTree abstract data structure, there is a remove() function. a. Explain briefly the purpose of the remove() function. b. The remove() function runs differently depending on the number of subtree on a node. i. Explain briefly, how to estimate the number of substrees given a binary tree node. ii. Give an example in a single sentence to justify that the worst-case time complexity for the remove() is O( n).arrow_forwardGive an inductive definition of the function nodecount(t) which computes the number of internal nodes in the binary tree t, where binary trees are defined inductively by the BNF equation t := *| N(t, t) Use this function to verify that nodecount(N(N(+, N(+,+), N(+, *)) = 4.arrow_forward
- Provide a nonrecursive binary tree implementation that keeps node data in a Vector. Element 0 of the data in this implementation refers to the root (if it exists). The left and right children of every non-null element I of the data are found at positions 2i + 1 and 2(i + 1), respectively. The parent of a nonroot node at I is thought to be located at position b(i 1)/2c, according to the inverse of these index relations. A null reference should be maintained for any data element that is not utilised to represent a node. avoid copying answers from any AI tools, as this will result in downvotes.arrow_forwardTl and T2 are two very large binary trees, with Tl much bigger than T2. Create analgorithm to determine ifT2 is a subtree of Tl.A tree T2 is a subtree of Tl if there exists a node n in Tl such that the subtree of n is identical to T2.That is, if you cut off the tree at node n, the two trees would be identical.arrow_forwardCreate a weighted quick union implementation for UF that always ties the shorter tree to the taller tree and follows the tree height tracking. Using your approach, establish a logarithmic upper constraint on the height of the trees for N sites.arrow_forward
- Recall that the height of a node in a binary tree is the number of edges (links) from the node to the deepest leaf in its subtree (e.g., the height of leaves is 0). To answer the following question, consider a complete binary tree T of size n in which the last level contains all possible nodes from left to right. (a) Specify the number of nodes at height h in T for any h≥ 0 (no proof is needed). (b) Specify the asymptotic running time of Bubble-Down for a node of height ʼn as a function of h. Then show the total time complexity to Bubble-Down all nodes at level h of T when we apply the Heapify operation on T. Justify your answer in one or two sentences. (c) Use your answer in part to prove that the time complexity of the Heapify operation is O(n). You need to sum the total work over all levels. Show your work. Hint: 0(1). x=0 =arrow_forwardSuppose an array is given A = [A, C, E, F, K, L, M, N, Y, Z] a. Draw a complete TERNARY tree from array A.b. Write preorder and postorder traversal for the created complete ternary.c. Draw the adjacency matrix and adjacency list for the complete ternary tree/graph. [You must consider the tree direction from top to bottom when drawing adjacencymatrix and list]arrow_forwardGiven a binary search tree (BST), find the lowest common ancestor (LCA) of two given nodes in the BST. According to the definition of LCA: "The lowest common ancestor is defined between two nodes p and q as the lowest node in t that has both p and q as descendants (where we allow a node to be a descendant of itself)." For example, in the figure from question 1, the LCA between nodes 5 and 46 is 21. You may use the following typedef structure. The function returns the reference of the node that is considered the LCA. typedef struct node_s{ int data; struct node_s * leftchild; struct node _s * rightchild; }node_t;arrow_forward
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