Hello everyone! I invented function F(x):{2,3,...}->N: if x is simple, F(x) is equal to its number in the sequence of prime numbers, if x is composite, F(x) is equal to its number in sequence of composite numbers. It is obvious that for any positive integer after a few uses of this function we will get 1. For example: F(22)=13, F(13)=6, F(6)=2, F(2)=1.

Now let's build an infinite binary tree: in the root is the number 1, for each vertex, which contains number N left son is N-th prime number and right son is N-th composite number. We'll get something like this: This tree has the following properties:

1. It contains all positive integers;
2. Parent of vertex with number N is F(N);
3. For each vertex the path from it to the root is a sequence of numbers generated by repeated calculation of the function F(x).

I don't know whether it might have some use, but it seems to me that it is not without a certain beauty. Besides, this, the same tree can be constructed in any two disjoint sets that their union is the set of natural numbers except 1 (if one of sets contains the number 1, almost nothing will change, just there will be loop from the root in himself, but it will not be a tree).

P.S. The same tree with depth 9