TAOCP 2.3.4.4 Exercise 7

Exercise 7. [**] [HM40] Obtain asymptotic values for the numbers of exercise 6. (See exercise 4.)

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Let $G(z)=\sum_{n\ge1}g_nz^n$, where $g_n$ is the number of distinct oriented binary trees with $n$ vertices. An oriented binary tree consists of a root together with an unordered collection of $0$, $1$, or $2$ binary subtrees. Therefore the argument leading to equation (3) applies with multiplicities restricted to $0$, $1$, and $2$. Hence

$$ G(z) = z\prod_{m\ge1}\left(1+z^m\right)^{g_m}. $$

Equivalently,

$$ \log\frac{G(z)}{z} = \sum_{m\ge1}g_m\log(1+z^m) = \sum_{k\ge1}\frac{(-1)^{k+1}}{k},G(z^k). $$

Exercise 4 shows that coefficients of generating functions defined by equations of this type have asymptotic behavior determined by the dominant real singularity. Let $\alpha$ be the smallest positive solution of $G(\alpha)=1$. Then

$$ 1 = \alpha\prod_{m\ge1}(1+\alpha^m)^{g_m}, $$

and the same analysis as in exercise 4 yields a square-root singularity at $z=\alpha$. Consequently,

$$ g_n = \frac{C}{n^{3/2}}\alpha^{-n} + O!\left(\alpha^{-n}n^{-5/2}\right), $$

where $C>0$ is determined by the local expansion of $G(z)$ at $z=\alpha$. Thus the numbers of oriented binary trees grow exponentially like $\alpha^{-n}$, with the universal $n^{-3/2}$ factor characteristic of tree-enumeration problems. ∎