Of the ten teams participating in the 2015 MLB playoffs, the Mets will have the most homegrown players, according to Jim Callis of MLBPipeline. Based on projected playoff rosters, the Mets will carry 15 players who were either acquired via the MLB first-year player draft or signed as an international free agent.
The closest team to the Mets in terms of homegrown players is the Cardinals, with 14 players, followed by the Royals, with 11. Conversely, the Cubs are projected to have only five homegrown players, with 15 players acquired via trade.
Not surprisingly, the Mets rank last in both free agent acquisitions and players acquired via trade, with three and seven players, respectively. Due to a need to limit payroll and a desire to develop talent from within, the Mets have built their team "the right way." That is to say, the Mets have used their farm system as a sturdy base for which to build a successful major league team, utilizing the trade market and free agency to supplement a strong core of homegrown players.
It should be noted that Callis' projection includes Juan Uribe, who has since been ruled out for the NLDS. If he is replaced by Kirk Nieuwenhuis, as some people are predicting, the Mets will add another homegrown player to their playoff roster. Nieuwenhuis was technically claimed off of waivers by the Mets after he was traded to Anaheim and subsequently released, but it is probably more in the spirit of this exercise to consider him a homegrown player.