  We show that an infinite Galton-Watson tree, conditioned on its martingale
limit being smaller than $\eps$, agrees up to generation $K$ with a regular
$\mu$-ary tree, where $\mu$ is the essential minimum of the offspring
distribution and the random variable $K$ is strongly concentrated near an
explicit deterministic function growing like a multiple of $\log(1/\eps)$. More
precisely, we show that if $\mu\ge 2$ then with high probability as $\eps
\downarrow 0$, $K$ takes exactly one or two values. This shows in particular
that the conditioned trees converge to the regular $\mu$-ary tree, providing an
example of entropic repulsion where the limit has vanishing entropy.
