I dont have public liabililty and I'm getting a handy man to cut some trees on my private property. Is there anything I can do or get him to sign to say he will be working at his own risk, basically preventing a potential claim?
Hire someone that has the correct insurance and ask for sight of the policy.
Your House Insurance would cover trivial work like mowing a lawn or doing any small odd job but where there are saws and in particular chain-saws involved, it's a different matter.