Height growth–rate at a given height: A mathematical perspective for forest productivity

•Novel mathematical approach to use as a site index in forest productivity.•Hierarchical structure considered by non-linear mixed-effects models.•Results highlight the height growth-rate at a given height is a potential site productivity variable. Understanding the height growth of trees is a fundamental component for scientific knowledge and management of forest ecosystems. The height of dominant trees at a reference-age, commonly known as the site-index, is the most widely used forest productivity indicator globally. Yet, it has been criticized for its restricted applicability to monospecific and even-aged forests, making it unreliable for natural forests or mixed-species where there is not a single meaningful age. Here, I develop a mathematical perspective for using height growth-rate at a reference-height as a new type of site index. I provide the mathematical basis for the proposed index and illustrate its application by fitting a nonlinear mixed-effects differential equation model to tree height growth data of three Nothofagus species in southern Chile. The proposed index allows us to foresee and analyze growth patterns, not only by representing growth-rates as a function of time but also of size. In doing so, the proposed index makes tree growth and productivity analyses accessible to a broader community of researchers.