I am trying to find a closed form formula for the following recursive function:
$$f_n(h)= \sum_{i=1}^{n-1} \binom{n-2}{i-1} \cdot (0.5)^{n-2} \cdot [ (f_{n-i}(h-1)\cdot \sum_{j=0}^{h-1}f_i(j)) + (f_{i}(h-1)\cdot \sum_{j=0}^{h-2} f_{n-i}(j))] $$
The base cases are the following:
$$ f_1(h)= \begin{cases} 
      1 & h=0 \\
      0 & otherwise
   \end{cases} \\
   f_2(h)= \begin{cases} 
      1 & h=1\\
      0 & otherwise
   \end{cases} 
$$ 
I have been trying to use the generating functions technique, but I have been unsuccessful so far and I was wondering if anyone has suggestions into how to solve this problem. 
Thank you for your help in advance

Edit:
I added the base cases