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Is there a known relation between the mass of a plant to the size of its roots?

For example, can I take the root size to be proportional to the biomass of the plant like in this relation: $$R = R_0 (1+EB) $$, where $R_0$ is the minimal size of the root (i.e. the diameter of the seedling), $B$ is the biomass of the plant, $E$ is some parameter, and $R$ is the length of a typical root?

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Your question specifically asks for an equation which gives root radius and your sample equation is for typical root length, neither of which I'm aware of. However, there is a known relation between the above-ground biomass and the below-ground biomass of a plant, which varies with species and environmental factors. This is the root:shoot ratio.

Excerpt from UQ website

This is a linear model where the constants represent differences in starting conditions (the intercept) and biomass partitioning (slope).

Roots and shoots are functionally interdependent and these two systems maintain a dynamic balance in biomass which reflects relative abundance of above-ground resources (light and CO2) compared with root-zone resources (water and nutrients). Whole-plant growth rate and root : shoot ratio are thus an outcome of genotype × environment interaction, but source of control is ambiguous.

The actual size of the roots - as in their extent - will depend on the soil characteristics, water table, the plant species, etc.

Source: http://plantsinaction.science.uq.edu.au/edition1/?q=content/6-3-1-biomass-distribution

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