There are two common ways of describing the topological properties of channel networks; one is to provide an exact description that gives all junctions and sources in sequence, and the other is to specify the Strahler stream numbers. For many purposes, the first method is too detailed and the second too broad. An intermediate method of classification, called "ambilateral," is proposed. The basic rule of this classification scheme is that two topologically distinct channel networks (TDCN) belong to the same class if one can be converted into the other by reversal of the right-left order at one or more junctions. Calculations based on an idealized model suggest that the ambilateral classification is more closely correlated with geomorphic and hydrologic properties than is the usual stream-number classification. Shreve (1966) has proposed that natural channel networks developed in the absence of geological controls are topologically random. This hypothesis can be tested by various statistical studies of the topological properties of a random sample of natural networks, including (1) distribution of individual TDCN for given N1, (2) distribution of ambilateral classes for given N1, (3) distribution of N2 values for given N1, (4) distribution of stream numbers in cumulative probability ranks for a given range of N1 values (a method employed by Shreve), and (5) distribution of the junction points of excess tributaries. Most of the tests listed above require that the sample of networks be selected without regard to order, a condition that is not satisfied by much of the published data. In order to obtain a suitable sample, 86 networks ranging in magnitude from 23 to 1172 were determined from stream systems in the United States. The statistical analyses generally support Shreve's hypothesis of topological randomness. One exception to this general remark is that the stream-number data for the western United States shows too many large (>4) bifurcation ratios; the discrepancy becomes more pronounced as N1 increases. Procedures are suggested for using observed deviations from the topologically random model to draw inferences about the presence of geological controls. © 1969, The Geological Society of America, Inc.