Safety factors are the nondimensional ratios between a characteristic of the present situation and the critical non-self-supporting one [9,46]. A safety factor of 1 (or lower than 1) means that the critical situation is reached. The higher the safety factor, the lower the risk. An important point to be assessed is whether the mechanical risk can be linked to material failure due to increasing bending or buckling because either could be limiting, but each requires distinct analyses that can lead to different conclusions. Bending occurs when a force component is acting perpendicular to the trunk, such as wind drag in a straight tree, or self-weight in a leaning tree. When bending stresses exceed the material strength, failure occurs. In a standing tree, the safety factor is then defined as the ratio of the material strength to the actual bending stress. Buckling is caused by a loss of stability of an equilibrium. For example, if a straight column is loaded under compression and at some critical point, the compressed equilibrium state becomes unstable, then any mechanical perturbation would induce a high degree of bending (see [7] for a more complete introduction).

In other words, the column is no longer self-supporting. Safety factors can be defined as the ratio of the critical weight to the actual weight. In plant biomechanics, interest is rather on what can be achieved for a given amount of aerial biomass. Safety factors for buckling are then usually defined as the ratio of the critical height to the buckling height, assuming relations, usually allometric, between weight and height. Mechanical models have been developed to calculate critical situations for both bending failure and buckling (e.g., [7,9,19,47-53]).

Such criteria are useful to compare the mechanical constraints between species or environmental situations. Many authors have also discussed the optimality of phenotypes at an individual (optimal stem taper) or population level (optimal stem slenderness), assuming that the optimal shape maximizes the height for a given diameter [19,48,50,54] or results in a constant breakage risk along the stem [47,53,55]. Slenderness rules, i.e., relationships that are usually allometric, between height and diameter within a population of trees are usually derived from the assumption of constant safety factors among the population (see [51] for a critical review and [56,57] for a general discussion about adaptative interpretations of allometries from mechanical and alternative hypotheses). However, several authors have discussed the values of safety factors when they are close or not to the critical limit, and their variability with tree ontogeny [21,49,58-60]. All of them found that safety factors against buckling decrease with growth in saplings as the competition for light became more intense and material resources that could be used for trunk growth become less available. A few authors have also studied safety factors in relation to species' shade tolerance and light conditions [58,61]. However, these approaches have always considered that trees have to avoid any critical situation and have never discussed the postcritical behavior of a tree nor the cost of height loss and its possible recovery. Nevertheless, buckling can lead to breakage or permanent, plastic stem lean, which is recoverable through the tree's gravitropic response (see Section 1.4.2). Breakage itself does not necessarily result in tree death and recovery can occur through healing of wounds or resprouting. Determining the conditions for buckling to occur is thus not sufficient, and the assumption that buckling is a catastrophic biological event remains to be tested in each particular case.

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