Right - it was the length of the string from nut to tuning post that needed the string tree. Or more precisely, the angle of the string. Going all the way didn't give enough downward force on the nut to let the string ring true - it would jump around in the slot. Arguably, even the staggered tuners of today may not give enough force.
So with a reverse headstock, you'll want the tree on the E and A string, potentially one on the D and G also, depending on the guitar.