  Driven by increased applications in biological networks and wireless sensor
networks, synchronization of pulse-coupled oscillators (PCOs) has gained
increased popularity. However, most existing results address the local
synchronization of PCOs with initial phases constrained in a half cycle, and
results on global synchronization from any initial condition are very sparse.
In this paper, we address global PCO synchronization from an arbitrary phase
distribution under chain or directed tree graphs. Our results differ from
existing global synchronization studies on decentralized PCO networks in two
key aspects: first, our work allows heterogeneous coupling functions, and we
analyze the behavior of oscillators with perturbations on their natural
frequencies; secondly, rather than requiring a large enough coupling strength,
our results hold under any coupling strength between zero and one, which is
crucial because a large coupling strength has been shown to be detrimental to
the robustness of PCO synchronization to disturbances.
