Go to Madison Junction in Yellowstone National Park. From here begins the west entrance road. If you are just entering the park from the west, the entrance road starts at the small town of West Yellowstone, Montana. Follow the entrance road to its exact halfway point. Here, you will find a bridge over the Madison River, called Seven Mile Bridge by the locals.
Park and look for trumpeter swans in the river either upstream or downstream of the bridge. You do not, however, need to park right next to the bridge; there are numerous turnouts along the river. Look for a bird that is almost entirely white, with the long, graceful neck that you would expect to see on a swan. In contrast with its bright white feathers, a trumpeter swan will have a deep black bill visible from a distance.
Be careful about how you behave around them. These birds are sensitive to human disturbance and you should give them more space than you would usually give an animal that is not particularly dangerous. Do this especially if the swans you see have young of the year. The young swans are called cygnets and you stand a fair chance of seeing some in the vicinity of Seven Mile Bridge.
Try Swan Lake next if you do not see any trumpeter swans on the Madison River. This lake got its name because swans do spend time there. It is in a broad meadow south of Mammoth Hot Springs in the northern part of Yellowstone National Park, just after the pass called the Golden Gate on the road to Norris Junction.
Try the Yellowstone River south of Canyon Junction. The road from Canyon to Fishing Bridge often runs within sight of the river. Again, look for strikingly large birds that are almost entirely white.
