Galton-Watson trees with vanishing martingale limit

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 $muge 2$ 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.