Give 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.

Structural Induction: Prove that a nonempty binary tree t of height h(t) has at most 2(t)+1 – 1 nodes.

Show the post-order traversal of the tree that results from starting with an empty tree and adding 67, 39, 12, 10, and 7. O 7, 12, 67, 39, 10 O 7, 10, 67, 39, 12 O 12, 67, 39, 10, 7 O 7, 10, 12, 39, 67

Consider the following function about binary trees: int dog( TreeNodeTypePointer p) { if (p==NULL) return 0; else if (p->left == NULL && p->right == NULL) return 0; else if (p->left != NULL && p->right != NULL) return dog(p->left) + dog(p->right); else return 1 + dog (p->left) + dog(p->right); } What does this function do? A. Computes the number of leaves B. Computes the number of internal nodes C. Computes the number of nodes with 2 children nodes D. Computes the number of nodes with 1 child node

Given a binary tree, find if it is height balanced or not. A tree is height balanced if difference between heights of left and right subtrees is not more than one for all nodes of tree. A height balanced tree 1 I 10 39 . 5 An unbalanced tree 1 I 10 I 5 Example 1: Input: 1 2 1 Output: 0

For a single binary tree whose in-order traversal is HDIAJTURP M S and whose post-order traversal is HIDJUTA PS M R a. Draw the binary tree.

A Binary Search Tree (BST) can be used to efficiently implement a sorted set. It stores uniquevalues and offers an efficient membership method (contains).A binary search tree (BST) is a binary tree with the additional binary search tree property.The binary search tree property states that for every node n in the tree1. n.data is greater than the data value of all nodes in the sub-tree rooted with n.left.2. n.data is less than the data value of all nodes in the sub-tree rooted with n.right.As we are implementing a set, all data (strings) in the tree will be unique. Every node in the treewill have its own distinct string. Strings will be compared using the canonical way (as defined bythe compareTo() method in the String class).For example, the binary tree on the left (below) IS a binary search tree and the one on the rightis NOT.You will implement several methods in the BinaryTree and BST classes. The BST class mustextend the BinaryTree class. Methods that you must complete (implement or…

Problem 3 Recall from lecture that a binary tree T is either: 1. null (the empty tree); or 2. a root noder with two subtrees T₁ and TR, both of which are binary trees. Consider the following algorithm numLeaves (T) that takes as input a binary tree T and returns the number of leaves in the tree. numLeaves (T): 1: if T == null then return 0 2: 3: else 4: 5: 6: 7: 8: TL, TR the left and right subtrees of T, respectively if T₁ == null and TR == null then return 1 else return numLeaves (TL) + numLeaves (TR) Prove by structural induction that for every binary tree T, numLeaves (T) returns the number of leaves of T.