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 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?

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 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?