There are some trees in the backyard of the hotel. Haven't been there since 2010 and I wasn't paying attention to trees back then, so I can't offer more. I think there's a running/biking trail near the hotel too that might offer sine possibilities. I don't know how the Japanese will view tree climbing. I suspect it will be extreme: either total apathy or a capital crime. I don't think getting your gear through security will be an issue. Korea, on the other hand, berated me for having a 4" crescent wrench, so don't divert to Incheon.