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We show that an infinite Galton-Watson tree, conditioned on its martingalelimit being smaller than $\eps$, agrees up to generation $K$ with a regular$\mu$-ary tree, where $\mu$ is the essential minimum of the offspringdistribution and the random variable $K$ is strongly concentrated near anexplicit deterministic function growing like a multiple of $\log(1/\eps)$. Moreprecisely, 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 particularthat the conditioned trees converge to the regular $\mu$-ary tree, providing anexample of entropic repulsion where the limit has vanishing entropy.

Galton-Watson trees with vanishing martingale limit