And now, we'll just talk about a few exercises to check your understanding of

singularity analysis. first one, and these are all web

exercises. they're on the book site.

so this one has to do with the standard function scale.

so and this is a context-free, but if you do symbolic transfer and solve then

you'll get a function from the standard scale that you can immediately get the

coefficient asymptotics for. so that's an exercise worth doing.

here's another one just to check the plug and chug for simple simple varieties of

trees. how many rooted order trees does every

node have zero, zero, two, or three children?

so enumerate the analytic combinatorics gives a quick way to get a estimate for

the number of such trees. and then as I discussed early on one

application of such a result is to know a lower bound on the number of bits that

you need to represent such a tree. so as usual read the corresponding

chapter in the text about singularity analysis of generating functions.

Again, there's a lot of deep water in there, so just try to get a feeling for

what's there not necessarily understand every detail.

and it's worthwhile to write up solutions to those two web exercises to check your

understanding the application of the, the basic theorems, the standard function

scale and a scheme like simple varieties of trees.

and then just for some more familiarity with the gamma function.

try plotting r and theta plots of 1 over gamma z in the, in the unit square and

see what that function looks like. those are some exercises to check your

understanding of singularity analysis and that's the lecture.