  For a bilinear map $*:\mathbb R^d\times \mathbb R^d\to \mathbb R^d$ of
nonnegative coefficients and a vector $s\in \mathbb R^d$ of positive entries,
among an exponentially number of ways combining $n$ instances of $s$ using
$n-1$ applications of $*$ for a given $n$, we are interested in the largest
entry over all the resulting vectors. An asymptotic behavior is that the $n$-th
root of this largest entry converges to a growth rate $\lambda$ when $n$ tends
to infinity. In this paper, we prove the existence of this limit by a special
structure called linear pattern. We also pose a question on the possibility of
a relation between the structure and whether $\lambda$ is algebraic.
