A non-associative arithmetic for shapes of channel networks

The purpose of this paper is to describe a method for analysis of one type of pictorial information that is abstracted from maps. The picture is a line diagram or graph that, in the language of graph theory, is a planted plane tree in which each vertex has a valency 1 or 3. In hydrology and geomorphology this type of graph is interpreted as a channel network that encompasses the topological properties of the network of rivers and streams comprising a drainage system. A recent survey paper by Dacey identifies a large number of properties of channel networks. Considering that many of these properties are clearly displayed by sketches of channel networks, the mathematical derivations seem unnecessarily complicated. This disparity in level of difficulty may reflect that the pictorial representation of a graph has a structure that is more amenable to analysis than does the conventional linguistic (i.e., mathematical) representation.