Answer: Weeping plum trees do exist, and are especially admired in Japan where they are featured in gardens. They are more unusual here, although you might see weeping Santa Rosa plum trees offered from time to time. It is quite possible and in my opinion more probable that your tree is a cherry of some kind. The reason I say this is that cherries and plums (along with peaches, apricots and almonds) are very closely related botanically and share the same primary Latin name, Prunus. Taken alone, this name certainly could mistakenly mislead one to assume that a cherry tree is a prune or plum.
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