I don't see any reason not to call them "subdivisions of claws," since that's exactly what they are; people working in subfactors [apparently][1] call them "star-shaped," or I guess in this case "claw-shaped." I don't know of any other name for them, though.

Now that I think about it, aren't these exactly the trees with exactly three leaves? Do trees with a specified number of leaves have a name?


  [1]: http://mathoverflow.net/questions/86/a-name-for-star-graph-with-long-laces