AFAIK that's one half of the story, but the other is something quite technical and cool in zoology that wasn't foreseen by ecologists at the time wolves were eradicated.

The wolves being removed caused something called meso-predator release, whereby the coyote numbers in the Yellowstone surged because of the cross over in prey species and lack of competition for that prey. BUT, that meant species not shared with wolves had their numbers heavily reduced by the increase in coyotes, mainly beavers.

When beaver numbers plummeted so did dams along the Yellowstone, and the the water level of the river went down and I think sped up, and that reduced areas for all animals and vegetative growth along the whole river. Reintroducing wolves caused coyote numbers to stabilise back at regular levels, beaver numbers did the same and the river stared to go back normal causing a huge bounce back across the board. Including reducing deer numbers back as coyotes don't hunt them so they did the opposite of beavers when wolves were removed for the same reason.... Meso-predator release but in the opposite direction.