Do all cats bring their owners "presents", especially birds?

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Have been thinking for a while about a getting a pair of cats, rather than kittens, probably from a re-homing centre.

But as dh points out, cats bring home "presents" and as he knows I'm not overly keen on rodents but I detest birds, to the point I would say I am phobic.

So, I need to know do all cats bring home birds? Do they do it a lot? Is there any way of choosing a cat who won't do this or a way of dissuading them?

My cat has never brought me a present - I live on quite a dense housing estate so there isn't a whole lot of wildlife around. We don't have a cat flap, so cat has to wait to be let in ( he goes out in the morning, then is let in at some point during the evening). If we saw that he had any presents with him, he certainly wouldn't be let in!