  Given a supercritical branching random walk $\{Z_n\}_{n\geq 0}$ on
$\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in
$[y,\infty)\subset\mathbb{R}$ at generation $n$. Let $m$ be the mean of the
offspring law of $\{Z_n\}_{n\geq 0}$ and $I(x)$ be the large deviation rate
function of the underlying random walk of $\{Z_n\}_{n\geq 0}$. It is known from
[6] that under some mild conditions, for $x\in(0,x^*)$, $n^{-1}\log
Z_n([nx,\infty))$ converges almost surely to $\log m- I(x)$ on the event of
nonextinction as $n\to\infty$, where $x^*$ is the speed of maximal position of
the branching random walk. In this work, we investigate its upper deviation
probabilities, in other words, the convergence rates of
\[\mathbb{P}(Z_n([xn,\infty))\geq e^{an})\] as $n\to\infty$, where $x>0$ and
$a>(\log m- I(x))^+$. This paper is a counterpart work of the lower deviation
probabilities [28] and also completes those results in [1] for the branching
Brownian motion.
