Tricky one - especially after the 'tree incident' Blatant cheek in my opinion!
Yes, unfortunately the solicitors route may be costly - however, Postdog's link is great - if you go onto this section of the site:
http://www.gardenlaw.co.uk/boundaries.html
it shows that you can obtain a copy of your neighbours and your own boundaries for £8 a pop (so might be a good starting point).
And Chrissa is correct, what can they do if you refuse? (It will cost them a lot of money in Sol's fees to take you to a small claims court)(maybe though, that's why they have begun the paper trail? Hmm...)
If it were me and my neighbours had already tried to cut down a tree illegally on my property, I would be tempted to say 'shove it' and see where it goes from there. It can get messy however, but if you already don't get on with them it's no great loss. I would just say I can't afford the cost at this moment in time, and every time they write you another 'demand' for money - They are trying to be clever by using the language as shown in their letter. But if they were really clued-up about the law and your need to contribute, surely they would not have leant over the fence to cut down your tree?
You could always say that you expect them to incur the full cost as compensation for the damage they did to said tree.
I am no expert on this though, so keep researching :-)