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I am trying to make a proof by induction of the following theorem.

If T is a full binary tree with i internal vertices, then T has i + 1 
terminal vertices and 2i + 1 total vertices. 

I have done this so far but I am just starting to understand proofs and am stuck about what to do next.

Base Case:
    P(1): 1 internal vertex => 1+1 = 2 terminal vertices

Induction:
    Assume true: P(n): n internal vertices => n+1 terminal vertices
    Show true: P(n+1): n+1 internal vertices => (n+1)+1 = n+2 terminal vertices

After this I am unsure of how to proceed.

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1 Answer 1

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SKETCH of proof: Let $T$ be a full binary tree with $i+1$ internal vertices. Let $v$ be a terminal vertex of maximal height, and let $u$ be the parent of $v$. $T$ is full, so $u$ has two children; let $w$ be the other child. Let $T'$ be the tree that remains when you remove $v$ and $w$ from $T$. Verify that $T'$ has $i$ internal vertices, and therefore by the induction hypothesis $i+1$ terminal vertices. One of these terminal vertices is $u$. When you restore $v$ and $w$ to $T'$ to recover $T$, you gain one internal vertex ($u$), you lose one terminal vertex ($u$), and you gain two terminal vertices ($v$ and $w$). The net change in internal vertices from $T'$ to $T$ is $+1$, as is the net change in terminal vertices, so $T$ must have $(i+1)+1=i+2$ terminal vertices.

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Verify that T′ has i internal vertices ... removing v and w turns u into a terminal vertex and thus reduces internals by -1 making i internals and i + 1 terminals? –  wazy Oct 4 '12 at 23:32
    
You are right, I thought for some reason about complete binary trees –  Belgi Oct 4 '12 at 23:52
    
@wazy: That’s right; and that means that $T$ must have had $i+2$ terminals. –  Brian M. Scott Oct 5 '12 at 0:30

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