On the optimal shape of tree roots and branches

Alberto Bressan, Qing Sun

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


This paper introduces two classes of variational problems, determining optimal shapes for tree roots and branches. Given a measure μ, describing the distribution of leaves, we introduce a sunlight functional (μ) computing the total amount of light captured by the leaves. On the other hand, given a measure μ describing the distribution of root hair cells, we consider a harvest functional (μ) computing the total amount of water and nutrients gathered by the roots. In both cases, we seek to maximize these functionals subject to a ramified transportation cost, for transporting nutrients from the roots to the trunk and from the trunk to the leaves. The main results establish various properties of these functionals, and the existence of optimal distributions. In particular, we prove the upper semicontinuity of and, together with a priori estimates on the support of optimal distributions.

Original languageEnglish (US)
Pages (from-to)2719-2762
Number of pages44
JournalMathematical Models and Methods in Applied Sciences
Issue number14
StatePublished - Dec 30 2018

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Applied Mathematics


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