Lyric discussion by Dressed2Depress 

Thats a really interesting insight, xpixiex, I hadn't considered the possibility of 'The Wild Rose' being used by newspapers etc as a name for the murder victim. Anyway, I think the firstverse sheds light on the notion of the murderer killing in order to preserve beauty:

From the first day I saw her I knew she was the one

[As early as the first ambigous line of the verse, Nick Cave introduces the motive of killing in order to preserve the image of beauty for eternity. In particular, he subverts the notion of 'the one' i.e. the one person with whom you spend the rest of your life with, by tying it in with the killers' predeliction for murdering beautiful women and selection of a victim.]

She stared in my eyes and smiled For her lips were the colour of the roses That grew down the river, all bloody and wild

[The comparison of her lips to bloody roses growing by the river bank is a premonition of what is to follow with the victim's murder.]

Also, in the third verse, the rose metaphor is tied into the notion of a flower representing female chastity:

On the second day he came with a single red rose Said: "Will you give me your loss and your sorrow" I nodded my head, as I lay on the bed He said, "If I show you the roses, will you follow?"

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