# 7.   Applications of Singularity Analysis

## VII.1

Use the tree-like schema to develop an asymptotic expression for the number of bracketings with $N$ leaves (see Example I.15 on page 69 and Note VII.19 on page 474)

## VII.2

Develop an asymptotic expression for the number of rooted ordered trees of size $N$ with no nodes having a single child.

## VII.3

For each of the following combinatorial constructions, give the appropriate schema for developing asymptotic enumeration results (simple variety of trees, exp-log, or implicit tree-like class). When two apply, pick the simpler one. $A = Z\times SEQ(A)$, $A = Z + SEQ_{>1}(A)$, $A = SET(CYC(Z))$, $B = Z\times (1+B)^4$, $A = Z\star SET(A)$, $A = Z + SET_{>1}(A)$, $A = SET(CYC_{>1}(Z))$.

## VII.4

(A. Ding) Show that the proportion of the 2-regular simple graphs on $N$ vertices whose cycles are all of length greater than $r$ is about $e^{3/4}/\sqrt{r}$ for large $r$.

## Program VII.1

Do r- and θ-plots of the GF for bracketings (see Web Exercise VII.1).