How do you keep track of the diplomas you've previously left?
You can tell because you can only take a diploma right when you place a painting that causes you to meet the requirements. So if you are placing a card with a tree on it, and it is your 4th tree, then you can see at that time that you must have chosen not to take the 3-tree diploma previously.
In other words, you don’t need a list of everything you have passed up previously; when deciding to take a diploma you can know based on the card you just finished playing which diplomas you are now allowed to take. Placing a 4th tree only enables you to take the 4-tree diploma; placing a card that has 2 trees on it, when you already had 2 other trees, would allow you to take either the 3-tree or the 4-tree diploma.