Fruit Bearing Trees

Hi.  Does anyone if it is possible to treat a tree in such a way that it does not bear fruit.  We have a pair of trees that, each year around this time, drop a small berry that tends to stain a walkway we recently constructed under the pair.  I have not identified the tree's genus/species.  It is however, very versatile--we have both bushes and trees that look to be the same plant.  The tree will also, just before the berries form, release very small (flowers?) that look similar to snow flakes.  In fact, for a period of one week, it appears to be constantly snowing, when we look outside the window under these trees--a strange sight in Los Angeles.
 
Thanks,
Doug
 
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