You will have a sturdier, healthier tree if you plant a new one, and it will be larger than the twig you have left, sooner. It will also look better than seeing the stump left over from the old one. If you don't completely remove the old stump, you will always have to look at it, the new tree will not fill-out the old stump. That part is dead. What is left of the old tree will fight to come back from the roots of the old tree, so if you have a sentimental attachment to the old tree, encourage the new start, and help it survive by removing the old stump and other old roots.
Trivia: industry studies show a 5-gallon tree from the nursery, and a one-gallon tree will be equal in size in 15 years. A start from the old tree will take about 1 year longer.
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