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If balancing the tree means better performance, would the end of a round (after death or victory) be a good time to balance the tree, or would it even make a difference in terms of performance during the match? I suppose it all depends on how hard one's bot is pushing up against the edge of the time slice it's allowed.

The end of the round probably would be a good time yes, though I've never really tried it. The re-balancing would take longer each round though, so it may make sense to only do that for the first few rounds of battle. As the number of rounds increases there would be diminishing returns to constantly re-balancing anyway.

The first few rounds makes sense.. Relatively quickly your dimension balancing should conform to the movement pattern of the enemy, and later rounds probably wouldn't shift the balance dramatically.

The distance dimension of a bot that heavily prefers a certain orbit distances will get properly split after just a few re-balancing operations, making further balancing on that dimension a comparative waste of time I would imagine.