  A good range of problems on trees can be described by the following general
setting: Given a bilinear map $*:\mathbb R^d\times\mathbb R^d\to\mathbb R^d$
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 $*$ to $n$ instances of $s$. When the coefficients of $*$ are
nonnegative and the entries of $s$ are positive, the value $g(n)$ is known to
follow a growth rate $\lambda=\lim_{n\to\infty} \sqrt[n]{g(n)}$. In this
article, we prove that for such $*$ and $s$ there exist nonnegative numbers
$r,r'$ and positive numbers $a,a'$ so that for every $n$,
  \[
  a n^{-r}\lambda^n\le g(n)\le a' n^{r'}\lambda^n.
  \]
  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.
