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Show: tex2html_wrap_inline12
Proof: Show: A complete binary tree of depth n has tex2html_wrap_inline53 n=0. Then the tree T has only one node and depth 0. Then the hypothesis holds, since tex2html_wrap_inline59 . Inductive Step. Assume that the hypothesis holds for all vaules tex2html_wrap_inline35 We demonstrate that hypothesis holds for k=n+1.
Assume a complete binary tree T has depth n+1. This tree can be viewed as containing two complete subtrees tex2html_wrap_inline69 and tex2html_wrap_inline71 with roots tex2html_wrap_inline73 and tex2html_wrap_inline75 respectively. The root r of T is joined to tex2html_wrap_inline69 and tex2html_wrap_inline71 by the arcs tex2html_wrap_inline85 and tex2html_wrap_inline87 , respectively. So, the number of nodes of T is the number of nodes of tex2html_wrap_inline69 plus the number of nodes of tex2html_wrap_inline71 plus 1 (for r). Applying the inductive hypothesis, since tex2html_wrap_inline69 and tex2html_wrap_inline71 are each of depth n, this number is

which completes the inductive step. Therefore, we can conclude by induction that hypothesis holds for all values of n.





Robert Morris
Mon Jan 19 15:08:23 EST 1998