now for exercises to test your understanding of this. remember I left for an exercise, applying the tree-like schema theorem to develop an asymptotic expression for the number of bracketings within leaves. That's Schroder's second problem. That's an interesting problem and you can read more about it on page 69 of the book and on page 474. the note on page 474 uses a different method to derive the result. so, you'll see for your exercise when you use the treelike schema that, you get out to the same result. and so that's a, a worthwhile problem to work to, understand how, how to apply treelike schema. in that, just that one exercise because, there were, others, in the last lecture which is the same week as this one. and again, take a look at the text, and again, particularly in this case, there is quite a bit there that I didn't cover in this lecture. Mostly around calculating the constant for context free and proving some of these theorems and schema, which is not what we are doing in this introductory treatment The next course in analytic combinatorics, we're going to talk about some of these books set and extensions. In write up solutions table, that exercise for solving numbers. and it's worthwhile as long as you have the code to do data plots from the last couple of lectures. Do our data plot of bracketing to see what that generate function looks like in complex plane. So those are the exercises for application of singularity analysis.