Recursive Branching Tree: Leonardo's Rule, Murray's Law and Sap Flow Resistance Lab

What Recursive Branching Is

Recursive branching is a pattern seen in nature, particularly in trees where branches split into smaller ones. This process can be mathematically modeled to understand how trees optimize their structure for efficient sap transport.

The simulation allows you to explore two key rules: Leonardo's area rule and Murray's law, which describe the relationship between branch radii at each fork.

Why It Happens

Leonardo's rule states that the cross-sectional areas of branches are conserved as they split. This ensures that sap can flow without significant pressure changes.

Murray's law, on the other hand, describes how branch radii change to minimize hydraulic resistance and energy costs for transporting sap.

How It Relates to Sap Flow

The simulation calculates sap speed per branch, hydraulic resistance, and the total pumping-plus-upkeep cost. These factors are crucial for understanding how trees manage their internal plumbing.

By experimenting with different branching rules, you can see which configuration is most efficient in terms of energy expenditure and flow rate.

Why Exponent 3 Is Optimal

Murray's law suggests that the ratio between successive branch radii should be (1/2)^(1/3). This exponent minimizes the total cost of maintaining the tree’s hydraulic system.

The optimal exponent of 3 is derived from balancing the energy required to pump sap against the resistance it encounters, making it the most efficient configuration.

Frequently asked questions

Try it live

Everything above runs in your browser — open Recursive Branching Tree: Leonardo's Rule, Murray's Law and Sap Flow Resistance Lab and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.