The trees in Figure 10.4.2 are identical rooted trees, with root 1, but as ordered trees, they are different.
![Two different ordered rooted trees](external/images/fig-ordered-rooted.png)
Case 0: Left subtree has size 0; right subtree has size \(n\text{.}\)Case 1: Left subtree has size 1; right subtree has size \(n - 1\text{.}\)\(\quad \quad \)\(\vdots\)Case \(k\text{:}\) Left subtree has size \(k\text{;}\) right subtree has size \(n - k\text{.}\)\(\quad \quad \)\(\vdots\)Case \(n\text{:}\) Left subtree has size \(n\text{;}\) right subtree has size 0.
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G1
. So the important thing about this first input is that it establishes z
as being a variable associated with power series over the integers. When the second expression defines the value of G1
in terms of z
, it is automatically converted to a power series.