We derive a new allometric scaling law for loopless networks, which we confirm with studies on rivers, exact network results and computer simulations. We provide evidence suggesting that ensemble averaging of the allometric property (where individual realizations are different networks) leads to results in excellent accord with the known limit scaling of efficient and compact networks with remarkably little scatter. Our results complement recent work suggesting that network-related allometric scaling in living organisms is regulated by metabolic supply-demand balance, because we show that scaling features are robust to geometrical fluctuations of network properties.