Can an insurer deny a claim for an act of mother nature?

Is it your insurer that is denying the claim, for your tree falling on the shed? If so, they can deny it AND you are also not liable for any damage, unless you were "at fault" in causing the tree to fall in some way. You are only liable for another's damage or costs when you do something wrong or are at fault in causing the damage; assuming that the tree was not known to you to be dead (and thus pose an obvious risk which it would be negligent, or unreasonably careless, to not remediate or correct), you were not at fault and so not liable. (The main time you can be responsible for a tree falling is when the tree is known to be sick, dead, damaged, leaning or excessively top-heavy, etc. and so poses an obvious risk; in that situation, you are expected to take steps to reduce that risk, and a failure to do so can make you liable.)
Your insurer is only responsible to pay for other people's damages or losses when you are liable, so if you are not liable because you had no fault--because it was an "act of nature"--they are not liable.
In addition, insurance policies are contracts: as with any contract, the insurer does not have to pay when the terms of the policy allow them to not pay. If the policy in question (whether yours or your neighbors) contained an exclusion for "acts of nature" or "acts of god" or the like, then that exclusion is valid--the insurer does not have to pay if the terms of the policy allow it to not pay in this circumstance.