Link to problem, it is not that hard to find the solution, but I can't quite understand why the operation done by the players can always be sorted into one of the three types of the following when the root is unknown:

• The first player sets the root in the first turn;
• The first player sets one of the leaves in the first turn, while the second player sets the root (That is only valid when at least 1 leaf is unknown);
• The first player sets one of the other nodes in the first turn, while the second player sets the root (That is only valid when the number of available nodes that is neither the root nor the leaf odd).

What I don't understand is that why can't one player choose the root or something else midgame, and why the condition for the third thing exists?