And again those are only the constructions that I've presented.

There's many, many other constructions available.

In those things, there's lots of possibilities.

They'll tell us something about the structure,

like a permutation is a set of cycles.

And actually lots of classic combinatorics can be dealt with in this way.

And then there's variations,

like generalizing the arrangements by adding another parameter.

The possibilities immediately become almost unlimited.

And not only that, when we get to a generating function equation,

we often have a universal law that will give us the asymptotics.

There'd be no way to go in and

get the exact result and then do asymptotics from the exact result.

In principle, you could do that because what

underlies analytic combinatorics is a bunch of very simple techniques,

but why would you if your goal is the asymptotic result?

So that's a very standard paradigm.

And also, combinatorial parameters can be handled,

and we'll see lots of examples of that.

And so that is, we're not just counting things,

we're counting properties of things.

But we talked about, in the generating function lecture, about the concept of

cumulating costs where rather than computing averages by using probabilities

what we do is we count up the total cost among all structures and then divide.

And it's reducing, finding an average for

a number expected in a random object to two counting problems.

So to find the leaves in a binary tree, we count trees using the standard

process to get to the estimate of the Catalan numbers.

But it turns out that the symbolic method works for

bivariate generating functions so that same construction will give

an explicit equation for the total cost.

So that's the leaves in all trees.

You just keep track of the leaves in another variable and

the same construction follows through, and then differentiate a value at a one to

get the leaves on all trees the way that we did before.

We're going to get again an explicit and then we have immediate transfer for

that one too.

So now we don't have to go into the detail,

we have these two asymptotic results and then we just divide.

And that's how we get to N over 4.

And again, we can do this without all the detail that we presented before.