your homeowners policy would cover any damage sustained to your property i.e. home, fence, shed, etc. after your deductible.  The tree &/or debris removal is maxed out usually at $1,000.00. Also, if you can prove that your neighbor was negligent had a dead  or sick tree and you advised them in writing certified return receipt at least 48 hours in advance and they ignored your notice and did nothing there is a good chance that your company can win a subrogation case against them.