The diode shifts up or down, all at once, the detected voltages in the "staircase".
If you insert zero, one or more diodes where the D1 is, the input voltage to turn on the first LED will change accordingly (and it will change with the same amount for the following LEDs, too). The "distance" between each voltage step in the "staircase" remains the same, but the beginning voltage of the entire "staircase" moves.