Recursive relations in binary trees
I think you're just confused about the definition of "height".  You write that you expect $f(2)=4$, where $f(h)$ is the number of leaves when the height is&nbsp;$h$.  A full binary tree with four leaves looks like this:

     o
   /   \
  o     o
 / \   / \
o   o o   o


So, in this case, you're taking the height to be the number of edges between the root and a leaf, and a tree of height&nbsp;$h$ has $h+1$ levels of nodes.

But you also write $f(2) = 2f(2-1) = 2f(1) = 2$, implying that $f(1)=1$.  A full binary tree with only one leaf looks like this:

     o


However, that tree has one level of nodes, so it has height zero.  The height&nbsp;$1$ tree is

     o
    / \
   o   o


so we see that $f(2) = 2f(2-1) = 2\times 2 = 4$, as expected.
Perhaps you were thinking of the following tree, which has one leaf and height&nbsp;$1$, but isn't a full binary tree.

     o
     |
     o