So that gives the characteristic function Z plus W squared.

There's the system and S equals a half, R equals a quarter.

Then we can compute the derivatives in the alpha and immediately get out the

coefficient and asymptotics just as before.

So that's a transfer appearing for how implicit tree-like classes in binary

trees as a simple example. so, let's just look at a complicated

example. A famous complicated example leads to

what's called Schroder numbers. A German mathematician called Schroder in

the 19th century studies a lot of problems like this and this is one so,

bracketing of N items is a tree that's got N leaves.

So the black ones are the leaves here and no unary nodes.

So and there's lots of applications of this, that are described in the book on

page 69. so how many bracketing within leaves so

here's all the breacketing. Say wiht three leaves, you can have a

[UNKNOWN] three or you can have two in left or two in the right.

again no unary nodes all nodes are either degree 0 or the degree 2 or more in the

sizes the number of leaves so its the 11 of size 4 and so forth.

that's equivalent to the number of ways of parenthesize an n items.

in part one of the course we talked about correspondences between trees an

parentheses systems. an if you go back an forth between these,

these slides you can see the correspondence between, bracketings an

ways of parenthesizing in items. So here all four are in the same

parentheses system. down here we have three with one on the

right or One on the left and then three, and so forth.

so that's where Charter was actually interested, was parenthesizations.

there's many other ways to look, look at this.

and-or conjunctive propositions are trees where and and ors,

alternate or what's called Series-Parallel Networks.

All of these things are equivalent to Bracketings.

So they are found in all sorts of different applications in combinatorics.

So what does it look like from the standpoint of analytic combinatorics?

Well a bracketing is a node or a sequence of more than one bracketings.

it's interesting to note that that combinatorial specification is probably

the most succinct way to describe what a bracket is.

and that's often true in analytic combinatorics.

We're talking about suc-, succinct ways of specifying things, and it turns out

that those lead us directly to automatic ways to enumerate 'em.

but that certainly is an important important part of the science is what is

it that you're trying to count? You need to specify and want to do it

succinctly. And [UNKNOWN] constructions are not a bad

mechanism for that. so, as usual, that translates immediately

to the generating function equation, so we have to subtract of the size zero and

the size one. And we could go ahead and solve for that

and, and so forth. But, the implicit tree like classes says

you don't really have to go ahead and solve for that.

What you have is your function is expressed as a function of Z in itself.

That's all you need to see. and as soon as you see that, then you can

get the coefficient asymptotics. I'm not going to do the details for this

example. that's left for your exercise for this

week. I'll do a more complicated one.