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Binary Trees Problem
4. Binary Trees. [15 marks total Recall that a binary tree is defined...
Recall from Assignment 2 the definition of a binary tree data structure: either an empty tree,...
Recall from Assignment 2 the definition of a binary tree data structure: either an empty tree, or a node with two children that are trees. Let T(n) denote the number of binary trees with n nodes. For example T(3) 5 because there are five binary trees with three nodes: (a) Using the recursive definition of a binary tree structure, or otherwise, derive a recurrence equation for T(n). (8 marks) A full binary tree is a non-empty binary tree where every...
2. A complete binary tree is defined inductively as follows. A complete binary tree of height...
2. A complete binary tree is defined inductively as follows. A complete binary tree of height 0 consists of 1 node which is the root. A complete binary tree of height h +1 consists of two complete binary trees of height h whose roots are connected to a new root. Let T be a complete binary tree of height h. Prove that the number of leaves of the tree is 2" and the size of the tree (number of nodes...
(10 points) Recall that the set of full binary trees is defined as follows: Basis: A...
(10 points) Recall that the set of full binary trees is defined as follows: Basis: A single vertex with no edges is a full binary tree. The root is the only vertex in the tree. Recursive rule: If T1 and T2 are full binary trees, then a new tree T' can be constructed by first placing T1 to the left of T2, adding a new vertex v at the top and then adding an edge between v and the root...
Design a divide-and-conquer algorithm for computing the number of levels in a binary tree. In particular,...
Design a divide-and-conquer algorithm for computing the number of levels in a binary tree. In particular, the algorithm should return 0 and 1 for the empty and single-node trees respectively. Please provide the pseudocode for your algorithm. What is the running time of your algorithm in the worst case using O() notation? Design a divide-and-conquer algorithm for computing the number of levels in a COMPLETE binary tree. In particular, the algorithm should return 0 and 1 for the empty and...
///Program needs to write in prolog/// 6. A binary tree is either empty or it is...
///Program needs to write in prolog///
6. A binary tree is either empty or it is composed of a root element and two successors, which are binary trees themselves. In Prolog we represent the empty tree by the atom 'nil' and the non-empty tree by the term t(X,L,R), where X denotes the root node and L and R denote the left and right subtree, respectively. The following Prolog term represents the given binary tree below. T1 = t(a,t(b,t(d,nil,nil),t(e,nil,nil)),t(c,nil,t(f,t(g,nil, nil),nil))) d...
Can someone help with these two problems? The following binary tree contains the characters 'A' through...
Can someone help with these two problems?
The following binary tree contains the characters 'A' through 'G' in its nodes. List the nodes in the order that they are visited by: A preorder traversal. An inorder traversal. A postorder traversal. The binary tree in Problem 2 is a Binary Search Tree since for every node, all elements stored in the left subtree of the node are smaller than the element in the node and all elements stored in the right...
Let T be a proper binary tree. Given a node v ∈ T, the imbalance of...
Let T be a proper binary tree. Given a node v ∈ T, the imbalance of v, denoted imbalance(v), is defined as the difference, in absolute value, between the number of leaves of the left subtree of v and the number of leaves of the right subtree of v. (If v is a leaf, then imbalance(v) is defined to be 0.) Define imbalance(T) = maxv∈T imbalance(v). (a) Provide an upper bound to the imbalance of a proper binary tree with...
Given a binary tree, determine if it is height-balanced. For this problem, a height-balanced binary tree...
Given a binary tree, determine if it is height-balanced. For this problem, a height-balanced binary tree is defined as: a binary tree in which the depth of the two subtrees of every node never differ by more than 1. Use following Node class, no height is stored in the Node /** * Definition for a binary tree node. * public class TreeNode { * int val; * TreeNode left; * TreeNode right; * TreeNode(int x) { val = x; }...
Problem 2 [35 points (155 15)]: Let Ti and T2 be two binary search trees containing the same elem...
Problem 2 [35 points (155 15)]: Let Ti and T2 be two binary search trees containing the same elements. In this problem, you will show how to transform Ti into T2 (i.e. Ti is altered to now have the same structure as T2) through a series of rotation operations. (a) Define a binary tree to be a right-going chain if no node in the tree has a left child (in other words, the tree is a linked list formed through...
Recall from Assignment 2 the definition of a binary tree data structure: either an empty tree, or a node with two children that are trees. Let T(n) denote the number of binary trees with n nodes. For example T(3) 5 because there are five binary trees with three nodes: (a) Using the recursive definition of a binary tree structure, or otherwise, derive a recurrence equation for T(n). (8 marks) A full binary tree is a non-empty binary tree where every...
2. A complete binary tree is defined inductively as follows. A complete binary tree of height 0 consists of 1 node which is the root. A complete binary tree of height h +1 consists of two complete binary trees of height h whose roots are connected to a new root. Let T be a complete binary tree of height h. Prove that the number of leaves of the tree is 2" and the size of the tree (number of nodes...
(10 points) Recall that the set of full binary trees is defined as follows: Basis: A single vertex with no edges is a full binary tree. The root is the only vertex in the tree. Recursive rule: If T1 and T2 are full binary trees, then a new tree T' can be constructed by first placing T1 to the left of T2, adding a new vertex v at the top and then adding an edge between v and the root...
Design a divide-and-conquer algorithm for computing the number of levels in a binary tree. In particular, the algorithm should return 0 and 1 for the empty and single-node trees respectively. Please provide the pseudocode for your algorithm. What is the running time of your algorithm in the worst case using O() notation? Design a divide-and-conquer algorithm for computing the number of levels in a COMPLETE binary tree. In particular, the algorithm should return 0 and 1 for the empty and...
///Program needs to write in prolog///
6. A binary tree is either empty or it is composed of a root element and two successors, which are binary trees themselves. In Prolog we represent the empty tree by the atom 'nil' and the non-empty tree by the term t(X,L,R), where X denotes the root node and L and R denote the left and right subtree, respectively. The following Prolog term represents the given binary tree below. T1 = t(a,t(b,t(d,nil,nil),t(e,nil,nil)),t(c,nil,t(f,t(g,nil, nil),nil))) d...
Can someone help with these two problems?
The following binary tree contains the characters 'A' through 'G' in its nodes. List the nodes in the order that they are visited by: A preorder traversal. An inorder traversal. A postorder traversal. The binary tree in Problem 2 is a Binary Search Tree since for every node, all elements stored in the left subtree of the node are smaller than the element in the node and all elements stored in the right...
Problem 2 [35 points (155 15)]: Let Ti and T2 be two binary search trees containing the same elements. In this problem, you will show how to transform Ti into T2 (i.e. Ti is altered to now have the same structure as T2) through a series of rotation operations. (a) Define a binary tree to be a right-going chain if no node in the tree has a left child (in other words, the tree is a linked list formed through...
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