Hello and thanks for using Just Answer.
I am Al and am happy to assist you with your enquiry.
The starting point for any claim is to establish the damage was caused by the tree in question. If this is accepted by you, this is straightforwrard but if not, unless it is painfully obvious that the roots emanate from the neighbours tree, it may be necessary for your neighbour to obtain an experts report to confirm the position.
Having established the above your neighbour would then need to demonstrate that either you knew of the damage being caused and failed to do anything about it or you ought to have known that damage was likely to be caused by the tree. If your neighbour cannot show that you knew about potential damage before it occured or failing which that damage to his structures was only a vague possibility ,a claim will likely struggle. However if the tree is very close to your boundary risk of damage to her drive will normally be deemed foreseeable as the neighbour will normally be deemed in such circumstances to have ought to have reasonably known, that reasonable remedial expenditure may be recovered by her as the neighbouring owner.
You do, therefore, ned to try and resolve matters just as soon as you can.
If your neighbour indicates that she intends to make a claim against you, you are best to speak to your Home Insurance company to see if you are covered.
I hope this assists and sets out the legal position.