For $n\ge 1$, let $f(n)$ be the number of rooted complete (unordered) binary trees with $n$ leaves labeled from $1$ to $n$ ("complete binary" means that every vertex has either $0$ or $2$ children and "unordered" means that the we do not specify which child is the left child or the right child). Then it is well known (e.g., Example 5.2.6 of Stanley's Enumerative Combinatorics 2) that the exponential generating function of $f(n)$ is given by $$F(x) := \sum_{n\ge1} f(n)\, {x^n\over n!} = 1 - \sqrt{1-2x}.$$
Now fix a positive integer $r$ and for $n\ge r$, let $f(n,r)$ be the number of rooted complete binary forests with $n$ leaves labeled $1$ to $n$, and $r$ roots labeled $n+1$ to $n+r$. By generatingfunctionology (e.g., Proposition 5.1.3 of Stanley's Enumerative Combinatorics 2), $$\sum_{n\ge r} f(n,r)\, {x^n\over n!} = F(x)^r = (1 - \sqrt{1-2x})^r.$$ This equation yields a formula for $f(n,r)$, and in fact we have:
Theorem. $$f(n,r) = {r(2n-r-1)!\over 2^{n-r}(n-r)!}.$$
The current proof I have of the Theorem simply notes that the generating function $F(x)^r$ coincides with a generating function for the Catalan tree, for which the coefficients are known to obey the above formula. My question is:
Is there a bijective proof of the Theorem?
