Removal of live trees is sometimes needed, but should be a last resort. Determine if the arborist is a member of the International Society of Arboriculture or the National Arborist Association. Membership does not guarantee quality, but lack of membership casts doubt on the person's professionalism. Ask for certification of personal and property liability insurance and workman's compensation. Then phone the insurance company to make certain the policy is current.
Low price is a poor gauge of a quality arborist.