Step 1: Parts
Their specs: (same for all colours)
forward voltage drop = 2.0V
max continuous current = 15mA
If you can, try to get LEDs that have the same specs - it's easier to design your tree.
The power supply used is from an old printer found out on the street - any sort of DC power supply will do, but the higher the voltage, the better. The one I am using supplies 30V, at up to 400mA. Enough power for 300 LEDs, but that's overkill. :-D