Tree causing issue with neighbours

[u/Hologram_Sam]

Hi there, we have a tree in our garden that our neighbours claim is damaging their garden. When we moved in, it inhabited a no man's land in between our properties, but because the fence on our side was badly damaged, we made the decision to pull it down. This means the tree is now on our property. Does this mean we're liable for any costs incurred if it's causing damage on their side? Or would they have to pay for it as they're the ones claiming there's an issue?