We analyse the shapes of star-shaped objects which are prealigned. This is motivated from two examples studying the growth of leaves, and the temporal evolution of tree rings. In the latter case measurements were taken at fixed angles whereas in the former case the angles were free. Subsequently, this leads to different shape spaces, related to different concepts of size, for the analysis. Whereas several shape spaces already existed in the literature when the angles are fixed, a new shape space for free angles, called spherical shape space, needed to be introduced. We compare these different shape spaces both regarding their mathematical properties and in their adequacy to the data at hand; we then apply suitably defined principal component analysis on these. In both examples we find that the shapes evolve mainly along the first principal component during growth; this is the 'geodesic hypothesis' that was formulated by Le and Kume. Moreover, we could link change-points of this evolution to significant changes in environmental conditions.

A full 3-D model was developed to simulate the elasto-mechanical behaviour of trees subjected to wind and gravitational forces with the aim of estimating the stress and strain distribution at the surface of the stem. The model was adapted to geometry and material properties of a 64-year old Douglas fir tree. The results are comparable, on the whole, with those of a finite element model of this tree. Original stem and crown data, as well as the applied forces, were modified manifold in order to study their importance on the change in fibre stress and thus, on the safety reserve against stem breakage.

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ABSTRACT:In the spring of 2000, field experiments were begun on three 20-year old grand fir (Abies grandis) to influence the elasto-mechanical behavior of the stem, as well as the distribution of the assimilate crown production. The aim was to analyze, and then describe and model the expected resulting change in stem growth. Three stem sections of one fir were reinforced by rigidly attaching T-shaped steel bars to reduce bending stresses. Preliminary calculations with an elasto-mechanical tree and force model (that had been developed for a different tree, but was adapted by taking the size differences into account) gave first indications for the experimental design in regards to the necessary number of the bars and their dimensions required to guarantee that the stem rigidity would be highly increased. Furthermore, the simulations proposed no increased risk of stem breakage in the non-reinforced stem parts. The stability of the second tree was decreased by hanging sand bags with defined masses on the branches. Directly after loading, a significantly changed swaying behavior could be observed, which should cause correspondingly higher stress in the fibers. As for this load case, the simulated results prognosticate only a negligible increase in stress. The roughly-adapted model used, is seemingly invalid for this tree. In regards to the third tree, the bark at the base of all branches of the eastern half of the crown was removed to prevent any import of assimilates into the stem. The branches were not cut off because the mass distribution and thus, the mechanical behavior, of the tree was to be influenced as little as possible. The experiment will be concluded in the winter of 2003, after a four-year growth period. In addition to detailed stem analyses, the spatial crown structure with its needle and branch mass distribution, as well as the mechanical wood properties of the stem will be measured. relationship between mechanical influence and growth reactions are hardly possible, per se. As a consequence, controlled thigmomorphogenetic experiments are demanded where growth and thus, the change of size and mechanical properties is registered and where applied forces are modified during the experiment in order to achieve a constant effectiveness (TELEWSKI 2000). Acting forces need not necessarily be constant, if they are continuously recorded as, e.g., in the experiments of BLACKBURN (1997). But such online strain measurements are expensive as they require a relatively sophisticated equipment and constant monitoring. Further, if the plot is open to the public, the risk of wanton damage must not be underestimated. Therefore, an alternative experimental design, which is not so demanding, is desirable and is presented in this work.In addition to mechanics, other factors such as the supply of assimilates have an obvious influence on secondary stem growth. Ever since PREßLER (1865) and HARTIG (1870), it has been know that the tree status which is defined by the size and the vigor of the crown determines the absolute secon...

We analyse the shapes of star-shaped objects which are prealigned. This is motivated from two examples studying the growth of leaves, and the temporal evolution of tree rings. In the latter case measurements were taken at fixed angles whereas in the former case the angles were free. Subsequently, this leads to different shape spaces, related to different concepts of size, for the analysis. Whereas several shape spaces already existed in the literature when the angles are fixed, a new shape space for free angles, called spherical shape space, needed to be introduced. We compare these different shape spaces both regarding their mathematical properties and in their adequacy to the data at hand; we then apply suitably defined principal component analysis on these. In both examples we find that the shapes evolve mainly along the first principal component during growth; this is the 'geodesic hypothesis' that was formulated by Le and Kume. Moreover, we could link change-points of this evolution to significant changes in environmental conditions.