Growth of bilinear maps II: Bounds and orders

A good range of problems on trees can be described by the following general setting: Given a bilinear map $\ast :{\mathbb{R}}^d\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ and a vector $s\in {\mathbb{R}}^d$, we need to estimate the largest possible absolute value $g(n)$ of an entry over all vectors obtained from applying $n-1$ applications of $\ast$ to $n$ instances of $s$. When the coefficients of $\ast$ are nonnegative and the entries of $s$ are positive, the value $g(n)$ is known to follow a growth rate $\lambda ={\lim }_{n\to \mathrm{\infty }}\sqrt[n]{g(n)}$. In this article, we prove that for such $\ast$ and $s$ there exist nonnegative numbers $r,r^{\mathrm{\prime }}$ and positive numbers $a,a^{\mathrm{\prime }}$ so that for every $n$, $$a{n}^{-r}{\lambda }^n\le g(n)\le a^{\mathrm{\prime }}{n}^{r^{\mathrm{\prime }}}{\lambda }^n\mathrm{.}$$ While proving the upper bound, we actually also provide another approach in proving the limit $\lambda$ itself. The lower bound is proved by showing a certain form of submultiplicativity for $g(n)$. Corollaries include a lower bound and an upper bound for $\lambda$, which are followed by a good estimation of $\lambda$ when we have the value of $g(n)$ for an $n$ large enough.