Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton-Watson trees. For example, let $mathcal{T}_1$ be the event that a Galton-Watson tree is infinite, and let $mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $mathcal{T}_1$ holds if and only if $mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $mathcal{T}_2$ holds if and only if $mathcal{T}_2$ holds for at least two of these trees. The probability of $mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a nonzero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterize the event undergoing the phase transition.
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