Not sure what you have there but it doesn't appear to be any 'tree'! At best it looks like suckers coming from the rootstock of what may once have been a cherry tree :-) As virtually all fruiting and flowering cherries are grafted, the rootstock is not considered a very desirable species and would probably not be something you want to encourage - small, sour fruit and meh flowers.

If you want a real cherry tree, purchase a named form of either a fruiting or ornamental cherry.

I'll defer to GG as usual, but I'm thinking this could also potentially be Prunus x cistena (Purple-leaved Sand Cherry), which can naturally grow as a suckering multi-stem shrub like this. I don't hear many people refer to them colloquially as "cherry trees" though.

The person who gave it to me told me what kind of cherry tree it is, but I forgot. I looked it up at the time and it was very common.