  We study the influence of the seed in random trees grown according to the
uniform attachment model, also known as uniform random recursive trees. We show
that different seeds lead to different distributions of limiting trees from a
total variation point of view. To do this, we construct statistics that
measure, in a certain well-defined sense, global "balancedness" properties of
such trees. Our paper follows recent results on the same question for the
preferential attachment model.
