Such sets are called *triods*. R. L. Moore (Concerning Triods in the Plane and the Junction Points of Plane Continua, Proceedings of the National Academy of Sciences, vol. 14, 1928) proved that every set of pairwise disjoint triods in the plane is countable.

https://www.pnas.org/doi/abs/10.1073/pnas.14.1.85