I have not seen a trend on what plants drop and those that do not. I have picked up arrowheads from plants, tending and clearing. I have also picked one up from fishing. I have had them drop from sequential plant tending and then gone without for many clicks. I think the issue is related to the quest sequence since the drops seem to be somewhat rare. At some point you will get a quest from the Duke where you will build a Beastie Laboratory and from there you can ask friends for arrowheads.

Similarly, I bought a practice dummy for 25k coins only to have it drop from another quest (freebie) a few days later as a reward. I have Sonja pending sausages for several days now...I just decided to wait it out and let the quests catch up because at some point I will have to build a butcher.

I've gotten 3 arrow heads while tending roots

you ask for them ...when your trying to build your silver arrows..there is a place to ask your friends for them

I would definitely go with Steven's suggestion. I bought a bunch of weeds and plowed 'em over. Requesting them from friends is a waste of a request. (Actually, ultimate wasted request is asking for cotton when you can ask for cotton thread. Haha.)

red tulip, goblins ears and roots give up arrow heads.

you can ask somehow for the arrow in the lab i guess as someone sent me a request for one