Answer: I am thinking here of the sprouts that shoot up at the base of the tree trunk. These are very difficult to move because they share the tree's roots. In most cases moving one of these will not give you a tree of the same type as the tree you have. The reason for this is that the tree is probably grafted, meaning a special top has been grown onto a sturdy rootstock. You should remove these suckers to help keep the tree healthy and directing its root energy to the grafted top rather than to the suckers.
With regard to fertilization, it is a good idea to run some basic soil tests and see how fertile your soil is (or isn't) and then fertilize based on the results. Your County Extension (432-2041) should be able to help you with the tests and interpreting the results. They should also have recommendations for routine care and maintenance for cherry trees in your area.
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