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Something no one has mentioned is the fact that large trees like that reduce energy costs in the summer by upwards of 25%, reduce heating costs in the winter by upwards of 15% Realestate costs also show that cutting down major trees like the ones you describe reduce property value by upwards of 15% the moment you cut them down. So if you want to really be practical about it, you would be really increasing your expenses by cutting them down. This is also on top of the cost of having someone cut them down in the first place. Seeing that this is a house that you don't yet own, I would suggest you keep searching. You will find something that fits your image of a perfect property elsewhere. All "treehugging" aside, if I was to look at it from a purely practical perspective, you would really be incurring a lot of extra expense by going ahead with a major change like that. Including bumping up your monthly costs for the foreseeable future.
In some places in the U.S. It's actually illegal to remove trees without getting city permission or at all. I know that Austin,TX, has strict laws about cutting cedar trees due to the enviromental wackos screeching about some endangered warbler bird needing the cedar trees to nest in and they have always been endangered in the first place. One rancher got fed up and cut down lots of cedar trees. Lo and behold! He had some springs come alive on his land that he had never known were there. The cedar trees were sucking up a massive amount of water....yet he got dragged into court by those wackos. Don't know if he won or not but it was ON his ranch land not on CITY land or limits!