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*Proposition:*
Let `T` be a tree of height `h` where every node has an outdegree
of at most 2. The number of tree nodes is less than
2^{h+1}:

forallT: (Tis tree /\forallxinV: outdeg(x) <= 2) => |V| < 2^{h+1}whereV=T_{0},h= height(T).

*Proof:*
Let `T` be a such a tree. We
proceed by complete induction on the height of `T`.

- Assume the height is
`h`= 0. Then |`V`| = 1 < 2 = 2^{h+1}. - Assume the height is
`h`> 0. Consequently the root of`T`has a child that is the root of a tree of height`h`-1 and possibly a second child that is the root of a tree of height less than or equal`h`-1. By the induction hypothesis, we thus have| `V`| <= 1+(2^{h}-1)+(2^{h}-1) = 2^{h+1}-1 < 2^{h+1}.

Author: Wolfgang Schreiner

Last Modification: January 26, 2000