The Brooklyn Botanic garden is a lifesaver for anyone who doesn't have the benefit of a country home (sigh, that's me).
It's tucked into the Brooklyn Museum's backyard and the nearby Prospect Park. It's divine at any time of the year. Highlights include the Japanese (Harumi?) Cherry Blossom Festival, June's rose garden explosion and fall's harvest.
The grounds are stroller friendly (but don't try to picnic, it's forbidden. You can only eat at the garden's cafe) and kiddies can roll around on the cherry tree lawn. The garden frequently offers free entry. Check website for more details and also look into kids programs.
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