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Unformatted text preview: n-1 moves are required to assemble a puzzle with n pieces. 6. Give a recursive deﬁnition of P m ( n ), the product of the integer m and the non-negative integer n . 7. A full binary tree is a graph deﬁned through the fol-lowing recursive deﬁnition. Basis step: A single vertex is a full binary tree. Inductive step: If T 1 and T 2 are disjoint full binary trees with roots r 1 , r 2 , respectively, the the graph formed by starting with a root r , and adding an edge from r to each of the vertices r 1 ,r 2 is also a full binary tree. Draw all full binary trees that can be obtain by ap-plying the inductive step at most 3 times (full binary trees of level 3). Use structural induction to show that n ( T ) ≥ 2 h ( T )+ 1, where n ( T ) denotes the number of vertices in T , and h ( T ) denotes the height of T . 2...
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- Fall '09
- Mathematical Induction, Natural number, Peano axioms, inductive step