Data Structures and Algorithm Analysis - Computer Science at ...

Sec. 5.8 Exercises 189techniques, see Managing Gigabytes by Witten, Moffat, and Bell [WMB99], andCodes and Cryptography by Dominic Welsh [Wel88]. Tables 5.23 and 5.24 arederived from Welsh [Wel88].5.8 Exercises5.1 Section 5.1.1 claims that a full binary tree has the highest number of leafnodes among all trees with n internal nodes. Prove that this is true.5.2 Define the degree of a node as the number of its non-empty children. Proveby induction that the number of degree 2 nodes in any binary tree is one lessthan the number of leaves.5.3 Define the internal path length for a tree as the sum of the depths of allinternal nodes, while the external path length is the sum of the depths of allleaf nodes in the tree. Prove by induction that if tree T is a full binary treewith n internal nodes, I is T’s internal path length, and E is T’s external pathlength, then E = I + 2n for n ≥ 0.5.4 Explain why function preorder2 from Section 5.2 makes half as manyrecursive calls as function preorder. Explain why it makes twice as manyaccesses to left and right children.5.5 (a) Modify the preorder traversal of Section 5.2 to perform an inordertraversal of a binary tree.(b) Modify the preorder traversal of Section 5.2 to perform a postordertraversal of a binary tree.5.6 Write a recursive function named search that takes as input the pointer tothe root of a binary tree (not a BST!) and a value K, and returns true ifvalue K appears in the tree and false otherwise.5.7 Write an algorithm that takes as input the pointer to the root of a binarytree and prints the node values of the tree in level order. Level order firstprints the root, then all nodes of level 1, then all nodes of level 2, and soon. Hint: Preorder traversals make use of a stack through recursive calls.Consider making use of another data structure to help implement the levelordertraversal.5.8 Write a recursive function that returns the height of a binary tree.5.9 Write a recursive function that returns a count of the number of leaf nodes ina binary tree.5.10 Assume that a given BST stores integer values in its nodes. Write a recursivefunction that sums the values of all nodes in the tree.5.11 Assume that a given BST stores integer values in its nodes. Write a recursivefunction that traverses a binary tree, and prints the value of every node who’sgrandparent has a value that is a multiple of five.