I think the proper term for the arms of Saguaro Cacti (Carnegiea gigantea) is branches, like a tree, but arms seems
more descriptive to me so that's what I think of them as. I also don't know what the proper term is for the
start of a new arm, but I think of it as an arm bud (lower left corner in the photo), so that's what
I'll call it here.
I've also been curious what the very start of a new arm bud looks like and saw one this day (upper
right corner of the photo). It looks like it splits out the skin of the cactus at one of the spine
pores (again, for lack of the proper term) and grows from there.
Photographed west of Tucson, Arizona USA.