Plant’s branch supporting bird (8)

I believe the answer is:
larkspur
'plant's' is the definition.
(larkspur is a kind of plant)
'branch supporting bird' is the wordplay.
'branch' becomes 'spur' (I am not sure about this - if you are sure you should believe this answer much more).
'supporting' means one lot of letters go next to another (in a down clue, some letters go below others).
'bird' becomes 'lark' (lark is a kind of bird).
'spur' put after 'lark' is 'LARKSPUR'.
Can you help me to learn more?
(Other definitions for larkspur that I've seen before include "Mediterranean plant with spikes of blue, pink or white flowers" , "Blue flower" , "Delphinium-like plant" , "Plant resembling a delphinium" , "Plant with flower spikes" .)
