Planar Binomial Coefficients

The notion of binomial coefficients $T \choose S$ of finite planar, reduced rooted trees $T, S$ is defined and a recursive formula for its computation is shown. The nonassociative binomial formula $$(1 + x)^T = \displaystyle \sum_S {T \choose S} x^S$$ for powers relative to $T$ is derived. Similarly binomial coefficients $T \choose S, V$ of the second kind are introduced and it is shown that $(x \otimes 1 + 1 \otimes x)^T= \displaystyle \sum_{S, V} {T \choose S, V} (x^S \otimes x^V)$ The roots $\sqrt[T]{1+x}= (1 + x) ^{T^{-1}}$ which are planar power series $f$ such that $f^T= 1+x$ are considered. Formulas for their coefficients are given.