Leah, I'm not sure how "good" an idea it is to plant a new tree in the hole (would any rotting tie up nitrogen or what-have-you? although that could be fixed by amendments), especially not the same tree (build up of diseases), HOWEVER, a lady I know promptly planted a flowering crab in the hole left when her old flowering crab was removed and it's done very well.

I would plant a new tree a little distance (I planted a plum maybe a yard or two from where a pear was removed) away and fill up that hole with good soil/compost and plant it up. That's mostly because it would require some work to plant a new tree right on top of roots from the old tree. If the area was shady, the rotting wood would make me think of woodsy areas and ferns and spring ephemerals would do well there. (My parents did that when they decided to remove a beautiful hemlock.) In a sunny area, I would find something and just plunk it in. Whether or not any wood/roots would tie up nutrients wouldn't make me worry, seems like that would be a good thing for the soil.

Have fun planting it up!
Kent