Under Connecticut law you can sue to recover the diminution of your property value or the value of the tree as timber. As the wood appears to have been left behind your choice is the diminution of value. You will need a real estate appraiser to testify as to the before and after value of the property.
Please note that as I read your post it seems obvious that the tree specialist was acting pursuant to his communications with your neighbor. So his acts are those of an agent and the principal, your neighbor is the one ultimately responsible for the damages.
Also your post implies that the tree was to be cut down in all events, in which case I don't see how you are damaged or have cause to "file a charge" about anything.
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