Nearly one-third of applicants for public-school pre-kindergarten programs were denied a seat for the fall, Department of Education officials said yesterday.

More than 8,800 families — or about 30 percent of the record-high 29,072 applicants — did not receive offers for full- or half-day slots for their tots.

But more than 2,200 seats were left unfilled because families specify only the schools at which they’re willing to enroll their kids.