We introduce gradient flow aggregation, a random growth model. Given existing particles \{x_1,\ldots,x_n\} \subset \mathbb{R}^2, a new particle arrives from a random direction at \infty and flows in direction of the vector field \nabla E where E(x) = \sum_{i=1}^{n}{1}/{\|x-x_i\|^{\alpha}}, 0 < \alpha < \infty. The case \alpha = 0 refers to the logarithmic energy {-}\sum\log\|x-x_i\|. Particles stop once they are at distance 1 from one of the existing particles, at which point they are added to the set and remain fixed for all time. We prove, under a non-degeneracy assumption, a Beurling-type estimate which, via Kesten’s method, can be used to deduce sub-ballistic growth for 0 \leq \alpha < 1, \text{diam}(\{x_1,\ldots,x_n\}) \leq c_{\alpha} \cdot n^{({3 \alpha +1})/({2\alpha + 2})}. This is optimal when \alpha = 0. The case \alpha = 0 leads to a ‘round’ full-dimensional tree. The larger the value of \alpha, the sparser the tree. Some instances of the higher-dimensional setting are also discussed.