JV> ... doing some clean-up yesterday, ... little knobs that are
 JV> left after a flower dies? Is that a rose hip?
Bingo!  
 JV> ...should I leave it on the stem or take it off? ...
It's your's... but if you are cutting your bush back, why would
you want to leave it?  It isn't productive to the bush,  but only
to the production of the *rose hip*...  not at all bad for you, but
the bush could care less ...   :)   Sometime before spring
flowering, you will want to cut that stem off anyway... so
unless you are wishing to produce 'hips', I say *cut it*!
cheers,
WL Sakowski
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