Preface: A bit of a rant! We enjoyed our visit but to be honest, even in early June the crowds were horrible plus the tour buses everywhere with tourists who made walking the boardwalks difficult and 'taking selfies' more important than actually viewing the natural sights around them. Getting a parking place at the geysers etc. sometimes involved waiting 15 minutes or more and dodging waiting cars on the main highways. If DH hadn't had a Handicapped placard we would have waited even longer. My last trip was in early August 2005 and far less crowded overall.
We did love our fancy dinner at Lake Yellowstone Hotel, our lunchtime picnics, seeing all the snow on the mountains, and viewing all the bison and elk who were everywhere we went! And please don't let our experience keep you or anyone else from making a trip in the future! Your trip could be very different from ours.