If a "cycle" is at least one each of a single, double, triple and homer in the same game, a "tricycle" is a baby cycle, one with three wheels -- only three of the required four events. There are four types of tricycles: a cycle that's missing a homer, one missing a triple, one missing a double, and the hardest tricycle of all, the one missing a single.
Jose Reyes has 5 homerless tricycles in Mets history, three of them in 2008. No other Mets has more than three homerless tricycles in his Mets career. In fact Reyes had three homerless tricycles in 2008 alone. The players with the most homerless tricycles in the major leagues since 1962, when the Mets began play, are Paul Molitor and Wade Boggs with 13 each, followed by George Brett and Willie Davis with 12 each.
David Wright has more triple-less tricycles as a Met than any player, with 13 (Mike Piazza had 12 and Carlos Beltran has 10). A-Rod and Barry Bonds lead the majors in triple-less tricycles since 1962, each has 35 in his career, ahead of Billy Williams and Juan Gonzalez with 32 each.
Double-less tricycles are unusual. Darryl Strawberry has the most in Mets history with 3, while Frank Thomas of the original '62 Mets had two within a week of each other in May of 1962. No other Mets has more than one, and there have only been 28 such games in Mets history. Across the majors since 1962, the most doubleless tricycles by any player has been 6, accomplished by Roberto Clemente, Andy Van Slyke and Ron Gant.
Finally, there have been only 6 single-less tricycles in Mets history. Amazingly enough, Gregg Jefferies has three of the six, two of them within a few weeks of each other in late August/early September 1988. The other three single-less tricycles in Mets history have been by Darryl Strawberry, Ron Swoboda and Joe Christopher. No player in the majors since 1962 has more single-less tricycles than the three by Jefferies -- Manny Trillo and Ellis Burks also had three and Brian Giles, who is still active, also has had three in his career.