Are you building this type of coop, or are you buying an existing coop from someone or somewhere else already built? It was my understanding that you intend to build from the design. If that is the case, you might want to consider dropping the extended roof top. I don't think you need that much extension for ventilation. Well, certainly not where I live. That might give you those extra inches you need not to trim the tree.
Are there any side windows or screens on that coop? If not, I might suggest building in some windows to allow cross ventilation and natural light into the coop. I bought some inexpensive garage type sliding windows for my coop which have worked out great for me. They have built in screens. I usually keep them open year-round, but I do close them temporarily if we get a blowing winter storm - just to keep out the snow.
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At the time, 3 years ago, I think I paid about $30 per window, but I imagine they have gone up in price. Anyway, I put one window on each side of the coop which provides some natural light and cross ventilation to freshen up the coop air.
A less expensive alternative would be using hardware cloth on a window frame (without glass), and just stapling on some plastic sheeting if you need to cover it during a storm.
You probably don't need extra ventilation or more natural light (said nobody, ever) but the time to consider it is before you build.