Under the assumption that the only information available on a drainage basin is its mean elevation, the connection between entropy and potential energy is explored to analyze drainage basins morphological characteristics. The mean basin elevation is found to be linearly related to the entropy of the drainage basin. This relation leads to a linear relation between the mean elevation of a subnetwork and the logarithm of its topological diameter. Furthermore, the relation between the fall in elevation from the source to the outlet of the main channel and the entropy of its drainage basin is found to be linear and so is also the case between the elevation of a node and the logarithm of its distance from the source. When a drainage basin is ordered according to the Horton-Strahler ordering scheme, a linear relation is found between the drainage basin entropy and the basin order. This relation can be characterized as a measure of the basin network complexity. The basin entropy is found to be linearly related to the logarithm of the magnitude of the basin network. This relation leads to a nonlinear relation between the network diameter and magnitude, where the exponent is found to be related to the fractal dimension of the drainage network. Also, the exponent of the power law relating the channel slope to the network magnitude is found to be related to the fractal dimension of the network. These relationships are verified on three drainage basins in southern Italy, and the results are found to be promising. ¿ American Geophysical Union 1993