Looking at how it is generated and given that it is indeed very, very large; why though is it not infinite?
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If you look at [this image](https://upload.wikimedia.org/wikipedia/commons/1/1e/TREE%283%29_sequence.png), you’ll see some intuitive things. First, once you’ve used the color green in tree #1, you can never use it again in any future tree without tree #1 being a subset of that later tree. In tree #2, you add the constraint that red can never be next to red again. In tree #3, you add the constraint that a red vertex can only have one child, and it must be blue. In tree #4, you add the constraint that you can’t have three blue in a chain without branches.
After only four constraints, it’s already getting really hard to construct trees, and over time you add more and more constraints. The question becomes “How many times can I chain sets of at most two blue vertices together, between separated red vertices, with each red vertex only having a single blue child, and… …without ever fully reusing a previous tree and also without using too many vertices?” And the answer is “a **lot**” but not “I can do this forever.”
Proving that you can’t do it forever is going to be beyond ELI5 unless there’s a simple proof I’m not aware of.
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