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Proof:
- Basis Step n=1 (trivial)
- Inductive Step. Assume that the hypothesis holds for all
vaules
We demonstrate that hypothesis holds for
k=n+1. I.e., that
- Therefore, we can conclude by induction that hypothesis holds
for all values of n.
Show: A complete binary tree of depth n has
n=0. Then the tree T has only one node and
depth 0. Then the hypothesis holds, since
.
Inductive Step. Assume that the hypothesis holds for all
vaules
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
and
with
roots
and
respectively. The root r of T is joined
to
and
by the arcs
and
, respectively.
So, the number of nodes of T is the number of nodes of
plus
the number of nodes of
plus 1 (for r). Applying the inductive
hypothesis, since
and
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