  Imbalance in the 3D structure of plants can be an important indicator of
insufficient light or nutrient supply, as well as excessive wind, (formerly
present) physical barriers, neighbor or storm damage. It can also be a simple
means to detect certain illnesses, since some diseases like the apple
proliferation disease, an infection with the barley yellow dwarf virus or plant
canker can cause abnormal growth, like \enquote{witches' brooms} or burls,
resulting in a deviating 3D plant architecture. However, quantifying imbalance
of plant growth is not an easy task, and it requires a mathematically sound 3D
model of plants to which imbalance indices can be applied. Current models of
plants are often based on stacked cylinders or voxel matrices and do not allow
for measuring the degree of 3D imbalance in the branching structure of the
whole plant.
  On the other hand, various imbalance indices are readily available for
so-called graph-theoretical trees and are frequently used in areas like
phylogenetics and computer science. While only some basic ideas of these
indices can be transferred to the 3D setting, graph-theoretical trees are a
logical foundation for 3D plant models that allow for elegant and natural
imbalance measures.
  In this manuscript, our aim is thus threefold: We first present a new
graph-theoretical 3D model of plants and discuss desirable properties of
imbalance measures in the 3D setting. We then introduce and analyze eight
different 3D imbalance indices and their properties. Thirdly, we illustrate all
our findings using a data set of 63 bush beans. Moreover, we implemented all
our indices in the publicly available \textsf{R}-software package
\textsf{treeDbalance} accompanying this manuscript.
