The old style Christmas tree lights with a twisted cable and all the lamps in series are notoriously frustrating to find faults with, but are very simply constructed.
Apart from checking the plug (fuse, connections, pins), cable and all the bulbs, there's not much else you can try.
This type of lights is known in the business as a class 01 appliance because it is neither earthed nor has double or reinforced insulation, and isn't recommended for sale or use in the UK any more, so you could use that as an excuse to get a new set of LED lights for next year.
Adam
January 2010