The situation between insurance and legal liability is different.
Typically a houseowner may have two different insurance policies - one for contents and one for his/her own property. They cover different things, and of course some people have both with the same company. The property insurance covers just that - your property; in many cases the shed, walls etc are not covered by this, but may be for example if subsidence affects both main property and shed/walls. If your tree falls over and damages YOUR property the property insurer may pay for repair to YOUR property but not neighbours.
Your contents insurance typically has a third party liability aspect - but it will depend on the nature of the policy. If the property owner has no contents insurance he has no third party insurance. It is this insurance that covers claims from subsidence from neighbours, just as it covers claims for trees falling over and damaging neighbour's property. It has nothing to do with forseeability or negligence and the policy may be invalid if negligence could be proven i.e. the tree owner should have done something and thus the insurer states that the owner is not insured for that peril.
As for legal liability, I once interviewed a property owner whose garage (& car) was demolished by an estate owned tree (large lime) that fell in the 1987 storm (i.e. the tree was on the edge of the estate and of little concern to the estate owner). The estate owner refused to have anything to do with it and told the old man to arrange for the tree to be removed by themselves, at their own expense. Such tree owners do exist! 3 days at least he was stuck in his house before he could get anyone to look at it.
A tree owner has not duty to invoke his insurance, but if he fails to inform an insurer of a risk and then later attempts to make a claim he is likely to be sent packing. So the above tree owner might think "act of god" - I can ignore, only to find a claim against him once the facts are gathered...which the insurance company will walk away from. So such an approach is risky.