We consider the Adversarial Queuing Theory (AQT) model, where packet arrivals are subject to a maximum average rate $0\leρ\le1$ and burstiness $σ\ge0$. In this model, we analyze the size of buffers required to avoid overflows in the basic case of a path. Our main results characterize the space required by the average rate and the number of distinct destinations: we show that $O(k d^{1/k})$ space suffice, where $d$ is the number of distinct destinations and $k=\lfloor 1/ρ\rfloor$; and we show that $Ω(\frac 1 k d^{1/k})$ space is necessary. For directed trees, we describe an algorithm whose buffer space requirement is at most $1 + d' + σ$ where $d'$ is the maximum number of destinations on any root-leaf path.