# A combinatorial interpretation for $n$-ary trees for negative $n$

The ordinary generating function $$T_n=T_n(x)$$ for the $$n$$-ary trees satisfies the functional equation $$T_n=1+xT_n^n.$$ This usually applies for $$n\ge 0$$, but the functional equation can be extended to negative $$n$$. Writing $$T_{-n}=1+xT_{-n}^{-n},$$ we obtain, by dividing through by $$T_{-n}$$, that $$T_{-n}^{-1}=1-x(T_{-n}^{-1})^{n+1},$$ i.e. $$T_{-n}(x)=\frac{1}{T_{n+1}(-x)}.$$ What would be a natural way to interpret this combinatorially? I.e. what are "$$n$$-ary trees" for negative $$n$$, why do we get the extra 1 degree, etc.

• I don't follow the transformations in the question, but from the functional equation it's quickly apparent that $T_{-1}$ gives Catalan numbers with alternating sign. Maybe there's a useful intuition based on anti-trees which contribute -1 to the g.f. coefficient? – Peter Taylor Dec 10 '19 at 23:29
• @PeterTaylor Just divide through by $T_{-n}$ and subtract the term with the $x$. I edited the question to include this now. – Alexander Burstein Dec 10 '19 at 23:38