Hi, welcome to GoY, to be quite honest there is no way to be sure which tree or trees are causing the problem as they are so close together, the soil is going to be a mass of roots, it could be the mature Laburnum, or the newer 1, or any of the others, there's just no way of knowing for sure, sorry, Derek.

It might be worth getting the advice of a tree surgeon - you will need one anyway to remove the tree and he maybe able to tell you which one or ones is causing the trouble by examining the area. As Derek says there's no sure way to tell without investigating.

You don't necessarily have to cut the offending tree (or trees) down. You can also do a root-trim with the help of a professional. This was done to my neighbor's giant Silver Maple that was encroaching upon my property. It didn't hurt the tree at all.

If your neighbour's building has been damaged by the roots of your trees, the only way they could know that for sure is if they've made an insurance claim - the insurance company should, in the circumstances, have taken root samples, which are then sent to a laboratory, to determine where/what the problem is, and when they do that, they can identify which roots they are, quite precisely. Therefore, your neighbour should be able to tell you exactly whether or not its a laburnum or any of the other trees causing the trouble. If they can't tell you that, that means they're just guessing that its your trees causing the issue - in which case, tell them you're fully prepared to cooperate and have any tree/s removed which might be causing the problem, but you need to know which tree - ask them to ask their insurance company to tell you, in writing, precisely which tree/s you need to remove in order to solve the problem. If no insurance company is involved, they need to get the root testing done themselves - its incumbent upon them to prove its your tree/s causing a problem rather than just assuming it is.

Obviously, if it turns out to be laburnum roots, then you'd need to have both laburnums removed.

The only option you have at the moment, if you don't want to insist on the official route, is to have all the trees removed that might conceivably be affecting their building - that'd be expensive and possibly entirely unnecessary. With so many trees there, its impossible to predict by looking which roots might be affecting their building, UNLESS one of the trees is much, much closer to their building than any of the others - and even then, its a guess.

Whether or not you choose to do away with your trees, you'll still have to deal with the root system below, especially where it affects your neighbor. You may be ordered to restore the damage if he goes the legal route.

I've counted at least 4 mature trees on your property, plus the lilacs. Just cutting down a single tree really won't make a difference, but give more space to the remaining trees to get stronger and fill out - and spread their roots far and wide.

The previous response isn't helpful at all, and is likely to increase your worry Galbnua9x - but try not to worry too much - if you are told which trees should be removed, that will inevitably mean the roots should be bored out to a depth of 18 inches anyway, if the tree surgeon called to do the job is aware of the problem and is told the roots need to go. It'll all get sorted, one way or another. The fact is, none of the trees you've mentioned are enormous trees, they're all standard garden trees, so you're not dealing with something like, say, 6 Sycamores (Acer pseudoplatanus), which would be infinitely worse with regard to root spread.

With regard to the Elder trees, though, if you do not have a gardener who attends regularly and who can cut those back at least once or twice a year, it's probably worth having those taken out anyway.

As I stated, the root system must be dealt with and the above comment simply reiterated my point anyway (thanks Bamboo)

Also, you may be prudent to deal with all the trees & roots to head off a repeat situation in a few years.