What claims could the British and Americans lay to the Oregon Territory?

Oregon Territory was originally claimed by Great Britain, who, under their claim, established successful trade outposts such as the Hudson Bay Company, etc. However, during the course of westward expansion (manifest destiny, whatever), after Lewis and Clark's exploration of Oregon Territory, many Americans settled the Oregon territory, including missionaries and fur trappers, who of course set up shop. In the end, the American claim was stronger, as you can see, Oregon isnt influenced by Britain like parts of Canada are now.