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It's well known that almost all graphs are asymmetric (have trivial automorphism group) and that almost all free trees are symmetric.

By which argument do I see whether almost all rooted trees are asymmetric or symmetric?

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Did you try adapting the argument for unrooted trees, e.g. as given in en.wikipedia.org/wiki/Asymmetric_graph#Trees? –  Pete L. Clark Apr 8 '11 at 14:26

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