I think this would be considered "your" line and the meter is at the house?
Aluminum would be normal here from the transformer to the meter socket, and from the meter socket to the service panel/main disconnect. I think it would be the thing to use for your extension too.
I wouldn't use a tree, I'm not sure that would pass various reviews such as insurance & electric code.
I would put in a pole ~midway, or hire it done by someone more likely, as serious digging and blasting may be involved. If you own the pole here, you can put a switch-controlled yard light on it, which could be handy.
You may need a mast and/or weather head at each end, plus something solid enough to screw the carrying cable anchor to, to do it correctly is my assumption. The minimum clearance to ground should be 14'. If you would ever run a communications line to the barn, I would allow for that possibility while you are at it.
I would run cable for a 100 amp service at a minimum. Considering all the variables, and what I know I don't know :laughing:, I would hire an electrician.