JV> I was doing some clean-up yesterday, and trimmed some of the 
 JV> deadwood from a rose bush. What are the little knobs that are 
 JV> left after a flower dies? Is that a rose hip?
Yep.
 JV> Whatever it is, should I leave it on the stem or take it off?
 JV> What's it do? I guess it isn't a seed?
It's the "fruit", and it does contain seed, if the rose isn't a
sterile variety as sometimes happens.  They should never even
form, you shoud be dead-heading the roses when the flower fades,
before the petals even fall off.  The BEST thing you can do for
your rose is to cut the flowers for the house.  This forces the
rose into more growth, more flowering.
... That's not a bug. That's a new feature
--- Blue Wave v2.12 [NR]
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