I start with a one-node tree. Then I repeatedly choose a node uniformly at random and add a child node to it, stopping when there are a certain number of nodes.
Treating all nodes as equivalent and ignoring the order of children of a node, I'd like to calculate the probabiity of producing a given shape of tree.
For example, there is only 1 way to produce the 1 or 2 node tree, so each of those are 100% likely if I stop at 1 or 2 nodes. If I'm producing a 3-node tree then I might get A->B->C or C<-A->B, each with 50% probability depending on which of A or B was chosen before adding C. From either of those trees, D could get inserted in one of three places (as a child of A, B, or C), but some of those results are equivalent:
1. A->B->C `->D 2. A->B->C `->D 3. A->B->C->D 4. A->B |->C `->D 5. A->B->D `->C 6. A->B `->C->D
1, 5, and 6 are all the same tree structure, so there's a 50% chance of producing that tree, and a 16.67% chance of producing each of the other three (2, 3, 4).