Hi Kirsten:
Thanks for your question -- I receive this exact question multiple times every week. The removal of a yard tree is a "service" and as such requires the skillset of a trained arborist for a fee to the homeowner.
I would venture a guess that 95-98% of arborists simply are not going to barter for potential lumber from a yard tree. The only exceptions are those arborists who are also woodworkers or custom furniture makers and the tree in question must be a desirable species and of unique value to that particular arborist / custom woodworker.
Sorry, but the removal of a yard tree (with rare exception) is a service that will require payment to a trained or certified arborist.