In Gilbert, not far from where Matthews grew up, the Big League Dreams Sports Parks features replicas of famous stadiums, including Wrigley Field, Fenway Park and Yankee Stadium, which makes it easy for major league dreams to take root. There are fewer than a dozen rinks in the Phoenix metro area, but the difficulty of procuring ice time did not deter Matthews. He excelled at both hockey and baseball, but there was never a question which sport he would pursue. His parents joke that they did not bother getting braces for Matthews because they knew his teeth were not long for his mouth.

“I think he loved baseball,” Brian Matthews said of his son, whose favorite position was catcher, “but there was too much standing around for him. If he could have batted every 15 seconds he would have loved it. Waiting around for the pitcher to throw the ball, it wasn’t active enough for him.”

Matthews was on the cusp of adolescence, he said, when hockey “kind of took over.” He explained: “I was missing a lot of baseball practices because I always wanted to be on the rink shooting pucks. So my parents could tell hockey was my No. 1 passion.”

For aspiring pro hockey players, the two main rinks are Oceanside Ice Arena, which is home to A.S.U., and the Ice Den, where the N.H.L.’s Coyotes practice. At both places, retired N.H.L. players come with the scenery. At youth games, the benches are alive with chalk talk from former players serving as volunteer coaches for their children’s teams. At the bantam game, Shawn McCosh, who played for the Los Angeles Kings and the Rangers, coached the Junior Coyotes. Assisting him was the retired goaltender Corey Hirsch.

Alex Hicks, another former N.H.L. player, who coached the Bobcats until he was hired as an assistant at A.S.U. in July, said: “The Southwest is an incredible spot for hockey players to develop. There are not thousands and thousands of kids playing hockey like in the traditional markets, but the kids who do play are passionate, and because of all the retired N.H.L. players who have settled here, I would say that per capita we have the best youth coaches in the country.”

With relatively few rinks, Arizona has a high concentration of coaches, including non-N.H.L.ers like Brad Bayer, Ron Filion and Boris Dorozhenko, who were instrumental in Matthews’s development, his father said. That is the upside. The downside, as noted by McCosh, who grew up outside Toronto, is that young players cannot while away the winters, as he did, working on their stick skills and stoking their creativity in their flooded backyards or on frozen ponds.