Endpoints of smooth plane dendroids

David Bellamy constructed in 1980 a surprising example of a dendroid with a connected set of endpoints. The example is smooth and lies in the plane. In this paper we prove that every smooth plane dendroid with one-dimensional endpoint set must contain a Bellamy dendroid (a smooth dendroid with connected endpoint set) or a Cantor fence (a copy of $2^\omega \times [0,1]$). In particular, if the endpoints are totally disconnected and one-dimensional then the dendroid is non-Suslinian. An example is constructed to show that this is false outside the plane.