Okay, I can see the recursive algorithm working provided the merge works correctly if you've got just 1 LCA.

I'm not entirely convinced that the merge for the 'counter' always works if you've got a single LCA though. It seems like it should obviously work, but I'm a bit concerned given that you can easily create a counter example where the formula doesn't work if you allow for multiple LCA. So somehow having a single LCA is vital for the formula to work and it's not entirely obvious to me why.

Anyway if you end up just calculating something equivalent to the union of the history (+some interpretation) then you're probably fine.