Posted: 24/07/2014 at 05:49
It depends what you want - if you're happy with a nondescript tree with green leaves which will probably get aphids but not have any blossom and fruit then leave it. However it will be hard to keep the other suckers from the roots down and just let the one grow.
I would dig it out, along with as much of the roots as possible, and dig in lots of organic matter to renourish the soil - it's probably pretty lacking in nutrients after having had a large plum tree growing there for 15 years.
I'd then plant a tree of my choice some distance away - it's not a good idea to plant immediately in the same place as one that has died as whatever killed it may remain in the soil and infect the new tree.