Exact and asymptotic results for the uniform random labelled tree on $n$ vertices have been studied extensively by combinatorialists. Here we treat asymptotics from a modern stochastic process viewpoint. There are three limit processes. One is an infinite discrete tree. The other two are most naturally represented as continuous two-dimensional fractal tree-like subsets of the infinite-dimensional space $l_1$. One is compact; the other is unbounded and self-similar. The proofs are based upon a simple algorithm for generating the finite random tree and upon weak convergence arguments. Distributional properties of these limit processes will be discussed in a sequel.