My wife and I live in a retirement village, so space is limited. The picture below is our Christmas tree which is about 900mm tall.
It is lit by 4 sets of seed lights, each powered by 3x AA batteries - a total of 12 batteries. Each string has it's own switch. 

Pictured is an original battery holder and the holder cut smaller.

I would like to replace these with a low voltage power supply and I have one that is 4.5V / 500mA.
I have basic understanding of circuit diagrams and if it was just one string, I am quite capable of connecting it.

My question is can I connect the 4 strings in parallel?
The current drawn by each set is very minimal, so a power supply at 500mA would appear to be almost overkill.
If it is doable, what would the wiring look like. i.e. do I wire the 4 cut down battery holders in parallel or do I just use one holder and wire the seed lights in parallel to that holder.

I hope that makes sense.

Thanks in advance

Graeme