The pipe model theory half a century on: a review.

Background: More than a half century ago, Shinozaki et al. (Shinozaki K, Yoda K, Hozumi K, Kira T. 1964a. A quantitative analysis of plant form - the pipe model theory. I. Basic analyses. Japanese Journal of Ecology B: 97-105) proposed an elegant conceptual framework, the pipe model theory (PMT), to interpret the observed linear relationship between the amount of stem tissue and corresponding supported leaves. The PMT brought a satisfactory answer to two vividly debated problems that were unresolved at the moment of its publication: (1) What determines tree form and which rules drive biomass allocation to the foliar versus stem compartments in plants? (2) How can foliar area or mass in an individual plant, in a stand or at even larger scales be estimated? Since its initial formulation, the PMT has been reinterpreted and used in applications, and has undoubtedly become an important milestone in the mathematical interpretation of plant form and functioning. Scope: This article aims to review the PMT by going back to its initial formulation, stating its explicit and implicit properties and discussing them in the light of current biological knowledge and experimental evidence in order to identify the validity and range of applicability of the theory. We also discuss the use of the theory in tree biomechanics and hydraulics as well as in functional-structural plant modelling. Conclusions: Scrutinizing the PMT in the light of modern biological knowledge revealed that most of its properties are not valid as a general rule. The hydraulic framework derived from the PMT has attracted much more attention than its mechanical counterpart and implies that only the conductive portion of a stem cross-section should be proportional to the supported foliage amount rather than the whole of it. The facts that this conductive portion is experimentally difficult to measure and varies with environmental conditions and tree ontogeny might cause the commonly reported non-linear relationships between foliage and stem metrics. Nevertheless, the PMT can still be considered as a portfolio of properties providing a unified framework to integrate and analyse functional-structural relationships.