Michigan contract law is based on Old English common law and definately recognizes oral contracts. The question at issue is whether a time frame (by the following Friday) was a material term of the contract. You state that it was, and that they failed to perform by the following Friday. At that time, the contract was breached by the removal service. They are entitled to the worth of the services they *did* perform, that is all.
If they agree to take a certain amount in payment, include "payment in full" on the check, or have them sign a written release. Do not pay cash!
Disclaimer: I am not licensed to practice in Michigan - consult a Michigan "contracts" or "business law" attorney directly.
Generally, verbal contracts are as binding as written contracts IF they have all of the necessary parts of a legal contract: (1) an offer to do something; (2) acceptance of that offer; (3) consideration for the offer; and (4) definite and certain terms.
It sounds like you had 1 - 3 above (offer to remove a tree; acceptance of that offer; consideration to pay for removal of the tree). But it sounds like 4 may be insufficient. While you agreed to price and the date of completion, you didn't discuss partial work, when payment was due, etc.
I think that this might not hold up as a binding contract, but you may wish to consult with an attorney in MI to see if there are any exceptions or cases similar to yours.
Verbal contracts are binding in Michigan. The tree removal service may or may not be entitled to partial compensation given that they performed some work. Please feel free to contact our office to discuss in greater detail.