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 USA, vol. 14, 1928, pp. 85-88) 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