The circuit is a basic "astable" circuit (which simply means it oscillates between states). In this case we use the venerable 555 timer integrated circuit to make a square wave oscillator with a frequency of a second or two.
The output of the oscillator circuit goes to 0V, stays there a while, then flips up to 9V, stays there a while then flips back to 0V and repeats again. When the output is 9V it lights up a set of LEDs. The "clever" bit is that when the output is 0V a second set of LEDs light up, because those LEDs are already connected to 9V (i.e. "positive") on their other side, and get connected to 0V (i.e. "negative") when the output of the oscillator goes down to 0V. The end result is that the two sets of LEDs flash alternately.
I found a small current flows even between the "high" output of the 555 and the 9V supply, so the LEDs are never completely off but flash from dim to bright (but in some ways that looks even nicer :o)
The parts needed for this project are
8 x identical "standard" LEDs
1 x "standard" LED for top of tree (can be same type as above, but you might want a different colour)
9 x 220 ohm resistors for the LEDs (typically 220 ohms but you might want to experiment with higher or slightly lower values to get the brightness levels you want. I used 100 ohm resistors on my red LEDs and a 470 ohm on the blue led)
1 x 555 timer IC (e.g NE555)
1 x 10M (10 mega-ohm) resistor
1 x 470k (470 kilo-ohm) resistor
1 x 10nF (0.01uF) ceramic capacitor
1 x 100nF (0.1uF) ceramic capacitor
1 x PP3 battery snap connector
1 x PP3 battery
Some connecting wire (e.g. 1 meter or less of thin equipment wire)
You can build this by poking the component legs through holes punched in the card, then soldering them on the back. This is kind of fiddly but it wouldn't be quite the same end result to build it on a board.
The following diagram shows the connections from the INSIDE of the card (i.e. the side you see when soldering). Click to enlarge
The LEDs must be inserted the right way round for this project to work. The anode (+) side of the LED is marked by a longer leg and the cathode (-) has a flattened edge to the lens. Use the + symbols next to the LEDs in the diagram above to make sure you put them in the right way round and remember the diagram shows the BACK of the circuit!
The resistors and capacitors have no specific polarity and can be soldered either way round.
Don't heat components for too long when soldering. LEDs are particularly sensitive to overheating and it probably won't do the 555 a lot of good either. It helps to bend leads together (so they stay joined by themselves) before soldering so you can be quick with the heat.
My card was particularly utilitarian, but I'm sure you can make it look much nicer :) sticking a sheet of card over the wiring hides the mess and strengthens it all. It would be good to see other peoples results!