Summary: We introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen-Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.