The remarkably regular geometric relations observed in stream networks have been widely interpreted as evidence of a distinctive structure that reflects particular geomorphic processes. These relations have also been interpreted as evidence that stream networks are topologically random, formed by the laws of chance. Neither of these inferences is justified. The oft-cited geometric properties are not specific to particular kinds of stream networks or to topologically random networks; instead, they describe virtually all possible networks. They therefore compel no particular conclusion about the origin or structure of stream networks.