I want to repair an LED Christmas light string with 3 wires per bulb. How is this string wired?
I took a string apart and traced the circuit.  It's interesting- pairs of LED's are in parallel and these pairs are wired in series.  The string uses a full-wave rectifier (improves flicker) and splits the diodes between the plug end and receptacle end (so both plug and receptacle are not too large).  So for an extra wire per bulb this design gains redundancy- one bulb of any or multiple pairs can be missing or open and the string still lights.  However if both bulbs of a pair are missing or open the string is dark.

Of course a parallel string is better for reliability but I think that would require four wires per bulb and a brick power supply of some type.

So as far as repairing it seems best to replace burned out bulbs immediately.  This is because if the other bulb paired with a burned out bulb goes then the entire string goes.  Then tracing which bulb is bad requires special effort.  Of course I've read that sometimes LED bulbs fail shorted instead of open.  So this helps.

Here is a schematic: