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The Game of Trees
In the game of trees, you start off with a single seed and then add branches. If somebody writes down a tree which contains a sort of a bit of a tree that's gone before, then the game's over. So how long could you keep going drawing trees without any previous tree being in it? Is the game guaranteed to end, for example? And if so, how long can it go on for? These are the kind of questions you can ask.