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**Unformatted text preview: **(a) Prove that the number of subtrees of a complete binary tree is not polynomial in the number of nodes. (b) Give an example of a class of trees { T n } where the number of subtrees is a polynomial in the number of nodes. 5. Let F i be the i th Fibonacci number (that is F (0) = 0 ,F (1) = 1 ,F ( n + 2) = F ( n ) + F ( n + 1)). Show that F n +2 = 1 + n i =0 F i . 6. Show that if you have a polynomial time algorithm for Hamiltonian Path, that you have a polynomial time algorithm for sorting. 7. The Bounded Degree Spanning Tree (BDST) problem is the following: Input: Graph G and integer k . Output: Yes, if G has a spanning tree where every node has degree at most k , No, otherwise. Suppose there is no polynomial time algorithm for Hamilonian Path. Show that there is no polynomial time algorithm for BDST....

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