HomeHomework HelpTextbooksTest PrepTextbook Answers
Zookal Homework Help

Homework Help

  1. Engineering
  2. Computer Science
  3. claim 2 for all n all full binary trees with...

Question: claim 2 for all n all full binary trees with...

Question details

Claim 2. For all n, all full binary trees with n internal nodes have height n -1 We have the following proof of the claim Proof. ?? By induction on n. For each n 2 1, let IH(n) be the statement all full binary trees with n internal nodes have height n -1 Base Case: Show IH(0). Every full binary tree with zero internal nodes is formed by case (A) of the definition, and thus consists of just a single leaf node. Therefore, every full binary tree with 0 internal nodes has only one leaf (which is also the root). Thus the longest root-to-leaf path has length 0, and IH(0) is true. Inductive Step: Let n 2 1 and show that IH(n) implies IH(n + 1). Let T be an arbitrary full binary tree with n internal nodes. Create the binary tree T having n +1 internal nodes by removing a bottonm leaf node in T and replacing it with an internal node connected to two children that are leaves. The longest root-to-leaf path in T is thus one greater than the longest root-to-leaf path in T, so the height of T is the height of T plus 1. Furthermore, by the inductive hypothesis IH(n), the height of T is n-1. Therefore T, has n + 1 internal nodes and height n-1 1 n, showing IH(n + 1). Identify and clearly describe the flaw in this proof (4 pts) Now go back to the proof of the first claim. Does it have a flaw (1 pt)? Clearly describe the flaw, or argue that the proof is correct (4 pts)

Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution