I have a function, fn that can take any number of inputs and produces one output. The order of the inputs doesn't matter.
However, if just one input is passed, the function just returns it as is, making this redundant. So to get something interesting, two or more inputs must be passed.
Now, I have $n$ objects on which to apply this function. I could either pass the first two as inputs to fn, get the result then pass the remaining $n-2$ as inputs to get the second result and finally combine the two to get the final answer. Or, I could pass all $n$ of them in one go and get the final answer. How many ways to get the final answer are there?
When we have three objects for example, there seem to be 4 ways to do this, forming four trees (see below). Also, is there a name for these kinds of trees?
You're looking at irreducible rooted trees with $n$ labeled leaves.... but more commonly, these are just evolutionary trees or phylogenetic trees with unlabelled interior nodes and $n$ labelled tips.
Enumerations and recurrence relations are given by J. Felsenstein ("The number of evolutionary trees", Systematic Zoology, 27 (1978), 27-33.) A scanned copy is available in the notes to OEIS A005373 (which is not, however, this sequence).
If I'm reading Table 1 correctly in the scanned paper, the number of such trees with $n$ nodes is $1,1,4,26,236,2752,39208,\ldots$ ... which leads us to the actual sequence of counts, at OEIS A000311.