  Leaf vein network is a hierarchical vascular system that transports water and
nutrients to the leaf cells. The thick primary veins form a branched network,
while the secondary veins develop closed circuits forming a well-defined
cellular structure. Through extensive analysis of a variety of distinct leaf
species, we discover that the apparently disordered cellular structures of the
secondary vein networks exhibit a universal hyperuniform organization and
possess a hidden order on large scales. Disorder hyperuniform (DHU) systems
lack conventional long-range order, yet they completely suppress normalized
large-scale density fluctuations like crystals. Specifically, we find that the
distributions of the geometric centers associated with the vein network loops
possess a vanishing static structure factor in the zero-wavenumber limit, i.e.,
$S(k) \sim k^{\alpha}$, where $\alpha \approx 0.64$, providing an example of
class III hyperuniformity in biology. This hyperuniform organization leads to
superior efficiency of diffusive transport, as evidenced by the much faster
convergence of the time-dependent spreadability $\mathcal{S}(t)$ to its
long-time asymptotic limit, compared to that of other uncorrelated or
correlated disordered but non-hyperuniform organizations. Our results also have
implications for the discovery and design of novel disordered network materials
with optimal transport properties.
