Grenflöden

Grenflöden are a way to describe how a conserved quantity moves through a branching network, such as a tree-like structure. The concept is used across disciplines to model how something entering a system is distributed among its progressively splitting branches. Common contexts include transport in biology, hydrology, and engineering.

In a rooted, directed branching network, each edge carries a nonnegative flow. At every internal node, the inflow is equal to the sum of the outflows, reflecting conservation of mass, charge, or another quantity. The split of flow at a node is characterized by branching ratios, which specify what fraction of the parent flow goes into each child edge. Leaves represent endpoints where the flow exits the network. By propagating the fractions from the root to the leaves, one can compute the distribution of flow among all leaves for a given input.

Grenflöden are often analyzed with linear flow models, using simple propagation of fractions, or with matrix methods that encode the network’s incidence and branching structure. They allow comparison of different network geometries, assessment of sensitivity to changes in branch splits, and calculation of effective flow properties such as leaf discharge, equivalent resistance, or funneling efficiency in more complex systems.

Applications span plant physiology, where grenflöden model sap transport in xylem or phloem; hydrology, where they describe discharge partitioning in river basins; electrical networks with dendritic structures; and information or supply networks where signals or goods are distributed along a branching path.

Limitations include assumptions of steady-state flow, non-interacting branches, and fixed splitting ratios; time delays and dynamic effects may require more detailed modeling.