The mechanical stability of the world’s tallest broadleaf trees

The factors that limit the maximum height of trees, whether ecophysiological or mechanical, are the subject of longstanding debate. Here we examine the role of mechanical stability in limiting tree height and focus on trees from the tallest tropical forests on Earth, in Sabah, Malaysian Borneo, including the recently discovered tallest tropical tree, a 100.8 m Shorea faguetiana. We use terrestrial laser scans, in situ strain gauge data and finite-element simulations to map the architecture of tall broadleaf trees and monitor their response to wind loading. We demonstrate that a tree’s risk of breaking due to gravity or self-weight decreases with tree height and is much more strongly affected by tree architecture than by material properties. In contrast, wind damage risk increases with tree height despite the larger diameters of tall trees, resulting in a U-shaped curve of mechanical risk with tree height. The relative rarity of extreme wind speeds in north Borneo may be the reason it is home to the tallest trees in the tropics.

Tree height growth is driven by the intense competition for light in young forest stands (MacFarlane

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Term Definition

Gravitational max height
The tallest a tree can grow without collapsing under gravity

Modulus of elasticity
A measure of a material's resistance to bending K The ratio of crown weight to trunk weight

Bending strain
Extension per unit length -in this case along a tree trunk

Wind-strain gradient
The slope of the best fit line relating bending strain to squared wind speed

Risk factor
The measured height of the tree divided by the maximum height consistent with safety, either from wind or gravity 18 Gravitational risk factors decrease with tree height within a forest canopy 20 We used two models to calculate the gravitational risk factors, the ratio of measured height to 21 theoretical maximum height each tree could reach before buckling under its own weight. The Euler 22 model (equation 1) represents a tree as a tapering cone with realistic trunk diameter and material 23 properties (Greenhill, 1881). The 'top-weight' model corresponds to a tapering cone with a top -24 weight to represent the crown (King and Loucks, 1978). A novel aspect of this study is the inclusion 25 4 of detailed 3D tree architecture based on TLS data. This allowed us to estimate the woody volume in 1 different parts of the tree and so estimate the weight of the stem and the crown separately. 2 Both models predicted a decrease in gravitational risk factors with tree height (Figure 2a). This 3 means that, as trees get taller, their radial growth is more than sufficient to compensate for their 4 increased height. We also calculated gravitational risk factors directly from continental height-5 diameter allometric equations (Feldpausch et al., 2011) and found the same trend (Figure 2b). The 6 intercept and exact shape of these risk factors depend on the relationship between tree height and 7 K, the ratio of crown weight to stem weight, but the negative trend and difference between 8 continents do not (S5). This result supports the hypothesis of King et al. (2009) who speculated that 9 understory trees have higher gravitational risk factors due to the prioritisation of vertical height 10 growth under intense competition for light, whereas overstory trees gain less advantage from 11 investing in height growth.  (Jagels et al., 2018). The red dashed lines at risk factor = 1 shows the point at which 19 a tree would be expected to buckle under its own weight. 20 Crown size, not material properties, determine gravitational stability 21 The risk factor estimates based on the two models followed a similar pattern (adjusted R 2 = 0.83) but 22 the magnitudes diverged substantially between models (Figure 2a). The 'top-weight' model 23 predicted that the trees are much closer to their gravitational stability limit than the 'classical' model 24 (increase in risk factor ranged from 0.062 to 0.581 with a mean of 0.214). This means that the 25 presence of a crown substantially decreases overall stability and demonstrates the importance of 26 crown dimensions in gravitational stability. 27 Comparing continents (Figure 2b) we find that trees in Africa and Asia (the latter data set being 1 dominated by sites in Borneo) have higher gravitational risk factors than those in South America. 2 These differences are driven by tree allometry, not by material properties. Clearly, trees experience 3 similar gravitational forces everywhere on Earth, and we suggest that these differences in 4 gravitational risk factors may be due to differences in wind regime. 5 Compared to changes in crown size, material properties (i.e. wood density and elasticity) had a 6 substantially lower effect on gravitational stability. Differences in risk factors using species-specific 7 or mean material properties ranged from -0.018 to 0.078 with a mean of 0.004. This small effect of 8 materials on gravitational stability is due to the fact that wood density and elasticity are strongly 9 correlated (Niklas and Spatz, 2010) and it is their ratio which affects gravitational stability. 10 Although its effect on gravitational stability is small, there is a consistent increase in the ratio of 11 wood elasticity to density with maximum tree height (Jagels et al., 2018). We found that including 12 species-specific material properties ( Figure 2a) tended to amplify the difference between short and 13 tall trees, with gravitational risk factors increasing for shorter trees and decreasing for taller trees. 14 Similarly, Figure 2b  Wind risk increases with tree height 17 The second aspect of mechanical stability is resistance to wind-induced snapping or uprooting 18 (Niklas and Spatz, 2012). Mode-of-death surveys show a wide variation the relative likelihood of 19 snapping and uprooting (Everham and Brokaw, 1996), presumably driven by site conditions, and a 20 survey in Danum Valley found that slightly more trees snapped than uprooted in this site (Gale and 21 Hall, 2001). We do not know of any field technique that can measure the risk of uprooting for large 22 trees in a tropical forest environment. We argue that on average the risk of snapping and uprooting 23 should be similar, as trees are unlikely to have evolved strong mitigation of one risk at the expense 24 of the other (e.g. excessive protection against uprooting would make little sense if tree snapping is 25 the dominant mechanical risk, and vice versa). Therefore, we quantified the (more empirically and 26 analytically tractable) risk of snapping and assumed that uprooting is, on average, equally likely. 27 We measured the local wind speed using anemometers attached to tall trees and measured the 28 bending strains at the base of the trees using strain gauges. As expected, Danum Valley is not a 29 particularly windy site and the maximum recorded wind speed in almost 2000 hours of data 30 collected between August 2016 to March 2017 was a 10 s mean of 7.2 ms -1 . We selected the 31 maximum wind speed and bending strain for each 10-minute window and calculated the wind-strain 32 gradient (Figures 3a, b). We then calculated each tree's risk factor for a given maximum wind speed 1 by comparing strain produced at that wind speed with the species-specific breaking strain (Table S1). 2 The absolute value of the wind risk factor depends on the chosen wind measurement type and the 3 aggregation window length. We therefore focus on relative risk factors.

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The wind-strain gradient was steeper for taller trees (Figure 3c) meaning that taller trees have a 14 higher risk of wind damage. This seems intuitive, since taller trees are exposed to higher wind 15 speeds. However, trees are known to adapt to their local wind environment through increased radial 16 growth (Bonnesoeur et al., 2016;Telewski, 1995). Therefore, it seemed possible a priori that these 17 adaptations could counterbalance the increased wind loading. However, our field data demonstrate 18 that this is not the case -the larger diameters of tall trees are not sufficient to compensate for the 19 increased wind loading. This is likely because the support costs scale with trunk circumference and 20 so increase disproportionately with tree height, making it increasingly difficult for a tree to grow 21 radially sufficiently to support the increased wind loading (Givnish et al., 2014). In addition, these 22 results show that variations in material property, such as an increased stiffness for a given density, 1 do not compensate for the increased wind loading. 2 Post-damage surveys have shown that tall trees are more likely to be damaged by wind than shorter 3 trees (Rifai et al., 2016;Silvério et al., 2018). However, these data represent the first direct 4 measurements of bending strains and mechanistic study of wind damage risk in a tall tropical forest. 5 Mechanical risk factor follows a U-shaped curve 6 We found no significant correlation between the wind and gravitational risk factors for the 11 trees 7 where field and TLS data overlap (p=0.17, 0.08 for the 'classical' and 'top-weight' models 8 respectively). This demonstrates that overall mechanical stability cannot be approximated by 9 gravitational stability alone. It should be noted that the wind and gravitational risk factors are not 10 simply additive, since the gravitational risk factors are derived from a model whereas the wind risk 11 factors are extrapolated from field data which necessarily include gravitational effects. However, for  Is Menara, the world's tallest tropical tree, close to its mechanical limits? 5 The world's tallest tropical tree provides an important case study for questions about mechanical 6 stability. Our analysis shows that the greatest mechanical limitation on this tree's height is likely due 7 to wind loading. Extrapolating from our field data to the height of this tree we estimate that it would 8 break at 24.1 ms -1 (measured at the location of the anemometer used in this study). However, this 9 tree, like the previous record holders from Malaysia (Jucker et al., 2017), is situated near the base of 10 a steep valley and will therefore be sheltered from the wind to a some degree. 11 In the above analysis we consider gravity and wind separately, but in nature they are combined. In in order to assess whether these tall trees are subject to mechanical constraints on maximum height. 9 We used TLS data to map the 3D architecture of tall trees and estimated the maximum possible 10 height consistent with gravitational stability. Gravitational risk factors decreased with tree height, 11 meaning that diameter growth more than compensated for increases in tree height, which is 12 consistent with previous work (King et al., 2009;Niklas, 1995). We also found that accounting for the 13 mass of the tree crown substantially reduced the maximum predicted height while material 14 properties had little effect. Our field measurements show that wind damage risk increases with tree 15 height, meaning that neither large diameters nor material properties counterbalanced this risk. 16 Therefore, overall mechanical risk factors follow a U-shaped curve with tree height (Figure 4). The 17 shape of this curve and the point at which mechanical risk shifts from gravity-dominated to wind-18 dominated depends on the local wind regime. 19 Overall, this result suggests that wind plays a role in limiting tree height, which has wide-ranging 20 implications including for forest structure and carbon stocks. Two features of geographical variation 21 of tropical tree height support a role for wind constraints. One is the observation that tree heights in 22 Amazonia and Africa are generally lower than in Borneo (Feldpausch et al., 2011). Northwest 23 Amazonia, in particular, has a wet and aseasonal rainfall regime similar to north Borneo (and hence 24 little expected seasonal soil water stress (Malhi and Wright, 2004)), yet much shorter maximum tree 25 heights. We hypothesise that, compared to the insular climate of Borneo, the continental climates of 26 Amazonia or Central Africa are likely to generate more intense convective events and downbursts 27 that lead to a higher frequency of extreme winds. Secondly, at a local scale, we note that the many 1 of the tallest trees in the Danum Valley area, including Menara, appear to be found in somewhat 2 wind-sheltered regions on the lee side of ridges (Shenkin et al., 2019), again suggesting that 3 maximum winds speeds are a limiting factor. The general decline of tree height with increasing dry 4 season intensity does suggest that hydraulic or carbohydrate supply is a constraint on maximum 5 height in many tropical forests, but in forests with little seasonal drought our analysis suggest that 6 rare maximum wind speeds may provide the ultimate constraint. 7 QSMs are available online. TLS scanning and data processing for Menara followed a similar protocol, 29 except that drone imagery of the crown was used to supplement the TLS data coverage using 30 structure-from-motion techniques (Shenkin et al., 2019). For this tree the QSM of the stem was 31 manually defined using a vertical profile of trunk diameter measurements taken directly from the 32 point cloud. Buttresses were removed from the tree-level point clouds prior to cylinder fitting to 33 generate the QSMs and a cylinder of the same height as the buttress attached to the bottom of the 1 QSM after fitting. The resolution of the point clouds decreased with tree height due to occlusion and 2 the divergence of the TLS laser beam. Therefore, the cylinder fitting process cannot detect the small 3 branches at the tops of the trees and the QSMs systematically underestimated tree height (Figure  4 S2). We therefore use height measurements taken from the point clouds for all analyses except for 5 Menara, which was directly measured by climbing. 6 As noted by Osunkoya et al., (2007) no generalizable definition of branching order is available in the 7 literature. We therefore manually defined the point at which the crown starts for each QSM (Figure  8 S3). We then calculated the ratio of crown mass to stem mass, K, and the positions of the centres of 9 mass for the crown and for the whole tree ( Figure S3). Previous work in similar forests found that 10 variation in crown architecture was primarily intraspecific, with only a small proportion of the total 11 variance explained by species (Iida et al., 2011;Osunkoya et al., 2007). These studies also reported a 12 decrease in relative crown size with tree height, which was fit with a power law function (Osunkoya 13 et al., 2007). Therefore, in order to generalize the risk factor calculation, we assumed a power law 14 relationship = 6.92 −0.69 , where the parameters were derived by fitting to the data using the 15 Matlab curve fitting toolbox (Mathworks, 2017) . This same relationship was used for all trees in 16 figure 2b so that comparisons between continents or across material properties gradients are not 17 affected by it. The intercept and exact shape of the risk factors depend on the details of this 18 relationship, but the negative trend, difference between continents and effect of material properties 19 do not (S5). It would be interesting to test for systematic variability in this parameter in future work. 20

Gravitational stability 21
The most relevant material properties for mechanical stability are the green wood density, , 22 modulus of elasticity, and modulus of rupture, . We collated material properties data 23 from the literature, see S1 for details. 24 We then calculated the theoretical maximum height each tree could reach before collapsing under 25 gravity, , using two different models. The 'classical' model (equation 1) of a tapering beam with 26 a circular cross-section (Greenhill, 1881) and the 'top-weight' model which includes a top-weight to 27 represent the crown centred at 0.9 (King and Loucks, 1978). The difference between the two 28 models is the value of the constant, C. In the top-weight model the weight of the crown is included 29 as per equation 2 which depends on the parameter K, the ratio between stem and crown mass. Wind-strain data collection and processing 3 Since there is no tower near this plot, anemometers were attached to tall trees (trees 1 and 6) 4 boomed out from the stem. This means that our wind data are site-specific and cannot be translated 5 to the standard 10 m above canopy height point of measurement. However, we found that this 6 unusual method proved highly useful (see S6). In order to measure the tree's bending in response to 7 wind we attached pairs of strain gauges to the stems of 19 trees at approximately 1.3 m. We used 8 three Campbell Scientific CR1000 data loggers and two CR23X data loggers to record the bending 9 strain data from October 2016 until April 2017. For the seven trees logged with CR1000 data loggers, 10 we collected hundreds of hours of data and found a clear bending strain signal (Figure 2a). We also 11 found clear signals for 9 of the 12 trees monitored with CR23X data loggers, although the data 12 availability was lower and the data were noisier. This complicated the uncertainty analysis and the 13 error propagation for the combined wind-strain gradient against tree height model. 14 The raw data consist of two mV readings per tree at 4 Hz and we calculated a single maximum strain 15 signal for each tree. This process involved (1) multiplying the raw strain signal by calibration 16 coefficients to transform from units of mV to strain; (2) bandpass filtering the data to smooth out 17 drift (3) re-projecting the signals onto into the North and East facing directions; and (4) combining 18 each pair of strain signals into a single maximum strain signal. We calculated the modal maximum 19 bending strain (Wellpott, 2008) and absolute maximum wind speed for each aggregation period (10 20 minutes or 1 minute). Using the maxima negates the need for a gust factor (Gardiner et al., 2000) 21 and focuses on the bending strains most relevant to wind damage. We then regressed maximum 22 wind against maximum strain and calculated the gradient. The wind speed measurement system 23 proved useful and wind-strain gradients did not vary much whichever anemometer they were 24 derived from (see S6 and S7). 25 26 We calculated risk factors in order to compare the roles of wind and gravity in maximum height 27 limitation. The gravitational risk factors were defined as the ratio of the measured height (based on 28 the point clouds since the QSMs systematically under-estimate tree height) to the modelled 29 maximum height, = / . We generalized gravitational risk factors using the continental 30 height-diameter allometries (Feldpausch et al., 2011) and, in the case of the top-weight model, the 31 power law relationship between K and H. We also used variation in material properties for tall and 1 short trees reported by Jagels et al., (2018) to estimate its effect on gravitational stability. 2

Risk factor calculations
In order to calculate wind damage risk factors, we extrapolated from field data to estimate what the 3 bending strain would be at a chosen threshold wind speed e.g. 20 ms -1 , 20 . The risk factor is then 4 , where is the breaking strain (S1). These risk factors are necessarily 5 relative to the chosen wind speed as well as the point of measurement and aggregating window 6 length. We therefore focus on relative risk factors instead of absolute risk factors. In order to directly 7 compare these results with other sites we would need data on the return time of extreme wind 8 events and wind measurements from towers approximately 10 m above the canopy. 9 Finite element analysis of gravity and wind Menara as the input, and simulated the effect of increasing wind speed both with and without 15 gravity (for details see S8). Simulating the effect of gravity proved difficult, since the trees were 16 obviously scanned in the presence of gravity and are therefore pre-stressed structures. The best 17 approximation to a proper treatment of gravity was to apply a reversed gravity force, export the 18 deformed positions of all the branches into a new analysis, then apply a downwards gravity force 19 and maintain it throughout the simulation. This gave us the desired effect of increasing the moment 20 due to self-weight as the crown deflects under high wind speeds. However, in the cases of other, less 21 straight trees, they failed to stabilize under the downwards gravity load and collapsed. This is likely 22 due to the fact that, in nature, trees develop asymmetric material properties to compensate for their 23 asymmetric architecture, but this variation in material properties cannot yet be included in the 24 The field data collected for this study are available at (https://doi.org/10.5285/657f420e-f956-4c33-b7d6-2 98c7a18aa07a). The field data processing scripts, summary data, TLS point clouds and QSMs are all available 3 (https://github.com/TobyDJackson/WindAndTrees_Danum). Radius and mass taper exponents were calculated 4 in Matlab (https://github.com/TobyDJackson/TreeQSM_Architecture). An updated library for analysing tree 5 structural information is also available in R (https://github.com/ashenkin/treestruct). 6 1 Howe, R.W., Hsieh, C.F., Hu, Y.H., Hubbell, S.P., Inman-Narahari, F.M., Itoh, A., Janík, D., Kassim, A.R.,