I've actually done this. I used an oil burner ignition transformer with a 17.5kV 45mA output. You probably don't need that much voltage though, you could probably do it successfully with a 2kV microwave oven transformer for a small piece. The trick is to soak the board overnight beforehand. This makes it easier for the current to flow between the terminals. The length of time you soak the wood and how porous/dense the wood is determines the voltage you'd need.
Here's a screen shot of the video description...my phone won't let me copy the text. Heed Clive's comments. People have died doing this, including an artist who created many works this way.
Like Ken, I used a 15 KV 30 mA Neon Sign Transformer. The downside? Yes, they can be dangerous and yes, they can kill you real quick. When I did this sort of stuff I was young and of course invincible.
But having found this thread i was wondering is AC needed or can this be done with a DC power supply? Reading around people anecdotally mention that you need AC so that the current path keeps being broken and new ones created ... and this creates the treeing. However Lichtenberg figures are created with DC discharges.
Most experiments i can find on youtube use MOT or neon sign transformers, so i'm just wondering if anyone has tried with DC?
My mrs is a keen woodworker and asked a while back about this, thinking of working with 2kv oustide with a wet sponge I said no way, thats asking for trouble.
However since then I've wondered about using an old tv lopty, which is a lot safer, still probably dodgy but I've been bitten my loptys and lived, so might give this a go, I have a lopty with a uc3842 or something like that as a lashed up eht supply somewhere.