Treliving said the process of finding a coach starts immediately.
The Anaheim Ducks and Ottawa Senators also have coaching vacancies, and the Minnesota Wild are undecided on the status of coach John Torchetti.
"Today's decision wasn't based upon anybody sitting on the on-deck circle," Treliving said. "Today is about Bob. This isn't about, 'Is there a prettier girl at the dance?' That's not it. You go through the process, you make a decision.
"Today, now, starts the process of finding who the next coach is, but this is not about who it could be, who it might be. Bob deserved to go through this thoroughly and a decision be made based upon what we feel is the best move going forward, not who might be out there."
Calgary, which finished third in the Pacific Division in 2014-15, allowed a League-high 257 non-shootout goals this season despite finishing 11th in shots-against per game (29.0). The Flames went from a plus-24 goal differential in 2014-15 to minus-28 this season.
Goaltenders Karri Ramo and Jonas Hiller, who played the majority of the games the past two seasons, struggled this season.
Ramo, who was placed on waivers and sent to the American Hockey League in October, went 17-18-1 with a 2.63 goals-against average and .909 save percentage in 37 games. He sustained a season-ending torn ACL in his left knee Feb. 11. Hiller had a 3.51 GAA and .879 save percentage in 26 games, the worst numbers of his NHL career.
"For us to move forward I felt this decision had to be made for what Bob can get from this group moving forward," Treliving said.
"Our special teams for a good portion of the year were 30th in the League. There's some style-of-play issues in terms of how to move forward. Our goaltending was not good. That falls on the general manager. I need to fix that. The way we play in front of the goaltender needs to be fixed as well. That lends itself to style of play to a certain extent. For those factors I felt a decision needed to be made."
The Flames were 22nd in the NHL on the power play (17.0 percent) and 30th in penalty killing (75.5 percent).
There were positives this season as the Flames' young players continued to develop.
Forward Johnny Gaudreau, 22, led Calgary and tied for sixth in the NHL with 78 points. Center Sean Monahan had an NHL career-high 63 points in his third season. Defenseman Dougie Hamilton, 22, had his best NHL offensive season with 12 goals and 43 points. Defenseman TJ Brodie, 25, had six goals and an NHL career-best 45 points.
"Each coach has their own style, their own way of doing things," Treliving said. "Bob, to me, is able to get a lot out of players. But when you sit back and review, and as I've gone through this process over the course of the past few weeks, I felt for us to move forward as an organization that it was time, it was important, it was critical that we made this decision."