Dynamic Programming, Binary Tree: How to factor in depth to recursion?
Turns out that I had a misunderstanding of the scope of dynamic programming.  As I understood it, I was "not allowed" to capture other things outside the recursion besides elementary operations.  

So, I just have to literally calculate the sum of values lower than the root on either side to factor in the depth: 

$$
T(i,j) = \sum_{i\leq l &lt; j}f_l + min\lbrace + T(i,k-1) + T(k+1,j)\rbrace: i&lt;k&lt;j 
$$

Where I can just calculate the sum outside of the recursion.