This particular tree isn't in New York - instead it's in South Jersey, near the beach. But no doubt, you've seen this plant around the city. The best spot in New York City - to me - to see redbuds, (Cercis canadensis) is on the path that leads up to the main loop of Central Park, from 5th Avenue and 85th Street. The redbuds there are planted on a slope so you are walking at eye-level with the canopy. There are also a lot planted around the NYU campus - in planters and in building courtyards.
There is a species of redbud in Europe commonly called the Judas tree. The folklore behind its name is that this tree was used by Judas Iscariot when he hanged himself, in shame for betraying Jesus. When he did this, the flowers on the tree (which once were white) turned pink with shame.
I can't find much about the derivation of the word Cercis. In Latin, Cerc- refers to a tail. Perhaps the tail shape of the small flowers? You can see the similarity of this flower with a sweet pea, which makes sense as they are both in the Fabaceae, or Pea, family.
People love redbuds for the bright pink flowers bloom all along the stems and branches of the trees, which is unique. But each spring, I was once again confused about when they bloomed. Sometimes they would bloom before the cherries, sometimes after.
Some background: All woody plants in a temperate region need a specific amount of daylight and minimum temperature to bloom. Some plants, like Forsythia, have no day-length requirement, and bloom almost immediately after a spike in temperature. Others, however, like redbud, need a certain number of days over a certain average temperature. Redbuds need 30 days of 50° plus weather before they bloom. This is probably one of the more rigorous thresholds for a spring-blooming plant and thus you see a lot of variation each spring as to when it blooms. It must be another reason people feel such sentiment towards the plant; when they bloom it means that spring is truly here and that the 40° days are behind us!