One time I heard a talk about "the" random tree. This tree has one vertex for each natural number, and the edges are constructed probabilistically. Connect vertex 2$2$ to vertex 1$1$. Connect vertex 3$3$ to vertex 1$1$ or 2$2$ with probability 1/2$\frac{1}{2}$. Connect vertex n+1$n+1$ to exactly one of vertices 1, ..., n$1,\dots, n$ with equal probability (1/n)$\frac{1}{n}$. This procedure will construct an infinite tree. The theorem is that with probability 1$1$, any tree constructed this way will be the same (up to permutation of the vertices).
My question is, does anyone know of a reference for this result? What is the automorphism group of this tree? Can anyone draw a picture of it?
I don't have any reason for knowing about this, just curiosity, and I wasn't able to turn up anything with a (not too extensive) internet/mathscinet search.
