Consider the structure of a rooted tree independent of its underlying set, (i.e. in the sense of trees as combinatorial species). I know a number of ways which we can encode any such tree in natural numbers, but all of them fail to be a bijective, i.e. Some numbers can not be decoded into any sensible tree structure. I would be very pleased to see such concrete correspondence. Thank you.
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See Y. Abe, Tree representation of positive integers, Applied Mathematics Letters Volume 7, Issue 1, January 1994, Page 57: