How can I prove that a complete binary tree has $\lceil n/2 \rceil$ leaves?
If the statement you're trying to prove by induction is


  For all positive integers $n$, a complete binary tree with $n$ nodes has $\lceil{n/2}\rceil$ leaves.


then the induction hypothesis must be


  For all positive integers $k&lt;n$, a complete binary tree with $k$ nodes has $\lceil{k/2}\rceil$ leaves.


Similarly, if the statement you're trying to prove by induction is


  For all positive integers $n$, a hoosegow with $n$ whatsits has $2^{\lfloor\sqrt{n}\rceil!}\cdot n^\pi$ nubbleframets.


then the induction hypothesis must be


  For all positive integers $k&lt;n$, a hoosegow with $k$ whatsits has $2^{\lfloor\sqrt{k}\rceil!}\cdot k^\pi$ nubbleframets.