Plan accordingly, A quick Google search indicates, over 4 million visitors last year, 2016, a record. Yellowstone is a remote section of Wyoming/Montana, with some pretty good accommodations. IMO 4 million visitors would tend to push the limits of the Park Services' abilities, IMO. Be careful out there. There has been recent discussion about limiting the number of visitors.
Note: Northeast in the park is Lamar Valley, much like Hayden Valley, with a lot of animals.
Note: The geography of this area is most interesting A very large lake, Yellowstone Lake, with the Yellowstone River flowing, north, eventually to the Missouri River, Mississippi, Gulf of Mexico. Those entering the park from the South would experience Lewis Lake, and the Snake River, flowing south to Jackson WY, eventually to the Columbia River and the Pacific Ocean. There is a continental divide between Lewis Lake and Yellowstone Lake. Weather is always changing, it can be very wet/cold and then dry/hot. Have fun
Note: The park is an active volcanic area, you will experience the multiple caldera's of at least three volcanos, that existed here. One of the eruptions/explosions, in this area, distributed ash over the entire United States. There has been discussion about recent increased volcanic activity.
Best wishes on your vacation trip
Mike CT.