## Introduction

It has long been thought that latitude should influence the vertical structure of forests, as a consequence of its effects on the angle of light that penetrates the canopy. Terborgh (1985), for instance, hypothesized that at higher latitude there is less vertical layering, and hence less diversity, because (1) the lower light elevation angles (angles of the light with the horizontal plane) cause light to pass through more leaves of the dominant canopy trees, and (2) trees at higher latitude have deeper crowns, because this is the form most effective in capturing light when light enters the canopy from a lower elevation angle. Modelling work has indeed suggested that, at low solar elevation angles, a canopy consisting of trees with deeper crowns is more efficient in terms of light interception and carbon gain, while at higher solar angles a shorter crown length produces a more efficient canopy (Oker-blom & Kellomaki, 1983; Kuuluvainen & Pukkala, 1987, 1989; Wang & Jarvis, 1990).

Although one can assume that more factors affect crown shape evolution at different latitudes, this modelling work has led to a long-standing notion in the literature that trees are indeed flat or shallowly domed near the equator while their shapes tend to be more vertically extended and steeply inclined closer to the poles, as a consequence of the effects of the solar angle (Richards, 1952; Halle *et al*., 1978; Whitmore, 1975; Terborgh, 1985; Hiura, 1998; King, 2005; Tateishi *et al*., 2010; Bomfleur *et al*., 2013, amongst others). However, most studies that refer to such a possible shift in crown shape base this argument on the theoretical work of Terborgh (1985) and Kuuluvainen (1992), and hence have implicitly assumed the general applicability of these models. This in turns means there is a heavy dependence on the model outcomes being robust against changes in the underlying assumptions; otherwise such translation from model outcomes to ecosystem functioning should be done more cautiously.

However, one of the main assumptions of these crown models may not be valid in naturally evolved forests. It is implicitly assumed that the benefit of a given vertical leaf distribution can be assessed via a population- or community-level outcome for a population in isolation, that is, through a simple optimization approach. This assumption does not apply to the evolution of crown shapes, because the benefit of height is pre-emptive access to light: taller individuals shade shorter plants but not vice versa (Ford, 1975; Weiner & Thomas, 1986). Therefore, the competitive benefits of height in general are frequency-dependent or game-theoretical (Givnish, 1982; Mäkelä, 1985; King, 1990; Falster & Westoby, 2003), meaning that the performance of individuals, and not of whole canopies, should be evaluated in competition with other possible strategies (e.g. Maynard Smith & Price, 1973; Riechert & Hammerstein, 1983).

It would therefore be interesting to determine whether such a game-theoretical approach would lead to the same predictions about how the solar angle influences crown depth. An important feature of game-theoretical models is that evolution through natural selection can lead to trait values that do not optimize the performance of the population as a whole (Schieving & Poorter, 1999; Anten & Hirose, 2001; Nowak & Sigmund, 2004; McNickle & Dybzinski, 2013); therefore, they predict different trait values from optimization models. But, as Franklin *et al*. (2012) showed, the two approaches could yield similar predictions. Yet, as they treat the selective nature of the environmental factors differently, their underlying mechanisms are not the same, and they cannot both give the correct interpretation as to why the shown patterns emerge (Anten & During, 2011). Different approaches become even more problematic when they produce results in opposite directions from each other. It is then that the conclusions drawn will depend on the model and its underlying assumptions, and will therefore not be robust against situations in which the conditions are not met.

Here, I show that a game-theoretical approach to model the evolution of crown depth leads to a different conclusion from the one prevalent in the literature, by showing that forests that experience light with a lower solar elevation angle are predicted to consist of individuals with shallower crowns than when the solar elevation angle is high. This conclusion follows from the following arguments.

- A higher crown base (
*Cb*) results in leaves that are placed higher up the canopy, resulting in higher light capture as a consequence of decreased neighbour shading. - The advantage of having a higher crown base, and hence a shallower crown, decreases as self-shading becomes a progressively larger fraction of all shading. This leads to an evolutionary endpoint (an evolutionarily stable strategy (ESS); Maynard Smith, 1982) where a change in crown base is no longer beneficial.
- As the relative amount of neighbour shading is greater at lower solar elevation angles, the advantage of increasing the crown base in terms of decreased shading by neighbours is also greater. Hence, the ESS is predicted to shift to higher values (i.e. shallower crowns) with decreasing solar angle.

In the model description I present the logic behind these arguments, supported by Fig. 1. Then, I shortly discuss the result of a simulation model (found in more detail in Supporting Information Notes S1) that was used as a sensitivity analysis to assess the effects of crown shape *per se*, and reduced light intensity with lower solar angle.