But would their reappearance really help?

She and her colleagues took a methodical approach to putting together available numbers on how deer populations grow, how many car accidents involve deer and how accidents increase with a growing deer population. They gathered information on how many deer a cougar might kill — about 259 over an average life span of about six years — and how much open, forested land was necessary for cougars to sustain a wild population — about 850 square miles.

Then the scientists tested a variety of mathematical models and came up with their projections. One of the questions they needed to consider was whether the cougars would be killing just deer that would die anyway from starvation or illness.

The scientists took what Dr. Prugh said was a conservative approach that about 75 percent of the deer the cougars killed would have died anyway. They also considered that as adult deer decrease in number, more fawns survive. So killing deer doesn’t immediately shrink the population.

Adrian Treves, the head of the Carnivore Coexistence Lab at the University of Wisconsin, Madison, who was not involved with the study, said he was impressed with the analysis and thought it might underestimate the benefits of cougars. He said in an email that there would probably be an even greater reduction in deer-vehicle collisions, “if governments and private citizens allow cougars to recover to historic levels.”

Figuring out the downsides of having cougars back was more difficult. The numbers of cougars would be considerable in some states once the deer and cougar populations stabilized — about 1,000 each in New York and Wisconsin, 350 or so in Missouri and only eight to 15 in New Jersey. They estimated lost livestock values in the areas studied at $2.35 million a year. But they were not able to get good estimates of pet loss, because it is hard to pin down which pets that disappear were killed by cougars. They may have been killed by coyotes, or cars, or may have been taken in by someone else.

Also, they could not account for the obvious emotional response to predators. Even if the estimate is correct that five times as many people would be saved by cougars as would be killed, death by deer and cougar are different.