Oh, I should add that functionally a constant voltage source is just a constant current source with a resistor in parallel. Also, functionally, a constant current source is just a constant voltage source with a resistor in series.
So basically, by putting a resistor in series with a constant voltage power supply and the LED, you're basically converting your source into constant current.
The reason you need to put a resistor in each branch is that since your LEDs drop a different forward voltage, your resistor takes up the slack and maintains the ideal current in each respective branch. If you only put one resistor in series before the LEDs, then you're going to limit the total current but not control the current in each loop.
Ideally, even if you had tightly binned LEDs, their characteristics will vary over time. Also, I believe manufacturers bin their devices with current and not voltage (as LEDs are current driven devices).