When the Nets headed into free agency last summer, they were coming off a sixth straight losing season, had virtually an empty roster and their most important player, Deron Williams, was set to enter free agency and be chased hard by the Mavericks, his hometown team. But Nets general manager Billy King eventually convinced Williams to stay, re-signed Gerald Wallace and Brook Lopez and traded for Joe Johnson, putting together the core of a Nets team that went on to win 49 games and make the playoffs for the first time since 2007. And though King did a very good job last summer, what he has done over the past few weeks — beginning with agreeing to the blockbuster trade with the Celtics for Kevin Garnett, Paul Pierce and Jason Terry on draft night last month — punctuated the Nets coming together over the last 13 months in ways even he didn’t think was possible. “No, I don’t think you could have dreamed of doing this as a GM,” King said recently. “You can dream of adding pieces, but not adding these pieces together.” The Nets have suddenly gone from a team that was a doormat just a year ago to one with as much star power as any team in the league after the additions of Pierce, Garnett and Terry, plus signing Andrei Kirilenko and retaining Andray Blatche. Those moves give the Nets tremendous depth and allow them to enter next season with realistic aspirations of competing with the defending champion Heat, Pacers and Bulls atop the Eastern Conference — while positioning them ahead of the Knicks in the battle for supremacy in New York.