Nonlinearity of root trait relationships and the root economics spectrum

The root economics spectrum (RES), a common hypothesis postulating a tradeoff between resource acquisition and conservation traits, is being challenged by conflicting relationships between root diameter, tissue density (RTD) and root nitrogen concentration (RN). Here, we analyze a global trait dataset of absorptive roots for over 800 plant species. For woody species (but not for non-woody species), we find nonlinear relationships between root diameter and RTD and RN, which stem from the allometric relationship between stele and cortical tissues. These nonlinear relationships explain how sampling bias from different ends of the nonlinear curves can result in conflicting trait relationships. Further, the shape of the relationships varies depending on evolutionary context and mycorrhizal affiliation. Importantly, the observed nonlinear trait relationships do not support the RES predictions. Allometry-based nonlinearity of root trait relationships improves our understanding of the ecology, physiology and evolution of absorptive roots.

I appreciate how the manuscript addresses some of the main methodological issues that have been raised to explain the lack of consistent RES relationships, i.e. the underlying phylogenetic properties of root traits, phylogenetic signals, the definition of absorptive roots and the effects of the species pool used. The approach and resulting outcomes are highly interesting.
My main concerns lie with the Discussion section. I am not sure whether I agree or disagree with the interpretation of the results, because this section is difficult to read and the argumentation hard to follow. I think the line of reasoning in the Discussion needs to be rewritten and perhaps restructured in order to be convincing -especially the results that do not agree with the hypotheses tested, e.g. the different trait relationships of woody and non-woody species, and the PIC analyses that lead to opposite relationships. I have listed larger and minor remarks in the attached Word document.

Monique Weemstra
Reviewer #2 (Remarks to the Author): In their manuscript "Non-linearity of root trait relationships and the root economics spectrum" the authors compiled a database of root diameter (RD), root tissue density (RTD) and root nitrogen concentration (RN) of absorptive roots of approximately 800 plant species. With this global fine root data they intended to resolve the so far mixed reports on the hypothesized tradeoff between resource acquisition and conservation traits according to the root economics spectrum (RES) and if this differs for woody or non-woody species. In contrast to the predictions of the RES the analysis revealed a non-linear negative relationship between RD and RTD. The authors convincingly argue that this relationship stemmed from the allometric relationship between stele and cortical root tissue which they substantiate with trait information on stele radius (SR) and thickness of tissue outside the stele (tToS) on 158 woody species and 13 non-woody species. In addition, the authors test this non-linear relationship along an evolutionary context and suggest a phylogenetic component in it for woody species with weaker phylogenetic conservatism of RTD than RD. Moreover, their results suggest that ectomycorrhizal woody species show stronger nonlinear relationships than arbuscular mycorrhizal species.
The paper is very interesting and the research questions are timely and very much discussed in the community of root ecologists, functional ecologists and disciplines related to biodiversity research. I also believe the topic of the root economic spectrum and its potential dependence on additional factors is of interest to a wider scientific community. However, this perception is of course stained by personal interests. The study is well composed and the paper very well written. The database is remarkable and data analysis is relatively straight forward and appears to be appropriate although the description could be more detailed in places.
My major concern lies with the originality of the results provided in this study. Most of the ideas, hypotheses and results given in this paper have been shown elsewhere already. Admittedly, this was on more limited data or plant groups or geographic regions but still the major outcomes of the paper are not really new. Even the very concept itself -nonlinearity of the allometric relationship -was published before by the authors (Kong et al. 2017). Results on the phylogenetic relationship of RTD, RD and RN were recently published by Ma and colleagues for roughly 400 species (Ma et al. 2018) and the relationship of phylogenetic structured root traits and mycorrhizal colonization in Valverde- Barrantes (2016Barrantes ( , 2017 . All these papers are correctly cited in the manuscript so I do not want to express here that the authors are not aware of this fact. Also, I am convinced that the more thorough dataset presented here fully warrants a new paper -I just think some of the wording could be better adjusted to the available knowledge.
The paper is based on a valuable database of root traits for more than 800 species. I am aware that the majority of this data is compiled from FRED yet substantial effort was put into additional data from other published literature. To my knowledge this database is not made publicly available or at least this is not stated in the manuscript or elsewhere in the accompanying material. I consider this fact a major flaw and would strongly encourage the authors to provide the compiled data e.g. to FRED for future use and ability of researchers to reproduce the work given the detail provided in this study.
Minor issues: Line 53: "These nonlinear relationships explain how sampling bias from different ends of the nonlinear curves produces conflicting trait relationships." I think this is a very important statement that is not really well highlighted in the rest of the paper. You do provide the data and kind of say this between the lines in the results and discussion, but I feel you could stress it even more that this is one potential cause for contradicting results in the field.
Line 73: I think reference 9 does not support the statement given here Line 101: Space missing after point Line 133: SR has not been defined before as abbreviation Line 182 ff: I am not convinced your data and analysis does properly separate the climate and mycorrhizal type argument so I would recommend to step a bit more careful in the wording here. As shown by one of your authors (Valverde-Barrantes et al 2017 New Phytologist) climate has a stronger effect on trait variation than mycorrhizal type. In your analysis you average trait values per species for different data origin and thus the climatic signal should be much blurred.
Line 218 ff: You refer to a model presented in the supplement. If I get this correctly your line of argument here is a model shows RD is important for root dry mass. From this you conclude that RD is more important than RTD without further mentioning additional tests and also that root mass and lifespan are positively correlated without reference. So you conclude that there is a relationship between lifespan and RD which had already been shown -so why the model? I do not get the point here I am afraid.
Line 224: I thought root foraging capacity is higher with lower RD not increasing with RD?
Line 226: The background argument of higher RN in thicker roots is already in the intro and repeated here, yet you make it sound like a new argument. Perhaps cut this down in the into than?
Line 229: which has been shown before. Sounds here like you show this for the first time.
Line 334: give more details on the linear mixed effect models.
Line 546: you say you use mean values per species for a trait (line 281) and that you have anatomical traits for 13 non-woody species. So why are there more than 13 points for non-woody species in these graphs? Do I miss something here?

Responses to editor and reviewers
Note: All the line numbers in our response letter refer to the revised version.
Editor's comments Your manuscript entitled "Nonlinearity of root trait relationships and the root economics spectrum" has now been seen by 2 referees. You will see from their comments below that while they find your work of interest, some important points are raised. We are interested in the possibility of publishing your study in Nature Communications, but would like to consider your response to these concerns in the form of a revised manuscript before we make a final decision on publication.
We therefore invite you to revise and resubmit your manuscript, taking into account the points raised. Please highlight all changes in the manuscript text file. We ask that you pay particular attention to the display and interpretation of your results, and provide further clarity in your argumentation. RESPONSE: We sincerely thank the Editor for considering our manuscript. In the revised version, we have paid particular attention to presentation and interpretation of our results and we have simplified and clarified our reasoning and argumentation. Further, we have taken into account all reviewers' comments. Together, we believe, this has greatly improved our manuscript. Our detailed responses are listed below.
Reviewer 1 I read the manuscript by Dr. Kong and co-authors with great interest. By assuming linear relationships among root traits, previous studies have reported inconsistent results on the relationships between root tissue density, diameter and nitrogen (N) concentration. This manuscript shows 1) that root trait relationships between diameter, tissue density and N% are nonlinear due to the allometric scaling of anatomical structures with root diameter, and 2) how this confounds our insights in root trait relationships, particularly regarding a root economics spectrum. Given the current interest in root traits in terms of resource economics and the debate on whether such traits are organized in a global root economics spectrum (e.g. 2 Ma et al. 2018, Nature) this is a novel, relevant and timely contribution to today's discussions in plant ecology.
I appreciate how the manuscript addresses some of the main methodological issues that have been raised to explain the lack of consistent RES relationships, i.e. the underlying phylogenetic properties of root traits, phylogenetic signals, the definition of absorptive roots and the effects of the species pool used. The approach and resulting outcomes are highly interesting. RESPONSE: We thank the reviewer for the positive evaluation and the constructive and helpful comments.
My main concerns lie with the Discussion section. I am not sure whether I agree or disagree with the interpretation of the results, because this section is difficult to read and the argumentation hard to follow. I think the line of reasoning in the Discussion needs to be rewritten and perhaps restructured in order to be convincing -especially the results that do not agree with the hypotheses tested, e.g. the different trait relationships of woody and non-woody species, and the PIC analyses that lead to opposite relationships. I have listed larger and minor remarks in the attached Word document.
RESPONSE: Based on the reviewer's comments, we have thoroughly reworked our manuscript, with particular attention to the line of reasoning in the Discussion section. For example, we have includes a new paragraph emphasizing the implication of the nonlinearity of root trait relationships on the specific root length (SRL): "Furthermore, the nonlinear relationship between root diameter and RTD advances our understanding of another key trait underlying the RES, i.e., specific root length (SRL, root length per unit root mass) 11,49 . Theoretically, SRL can be expressed as: SRL = 4/(π × RTD × root diameter × root diameter) 17,50 . If RTD is positively correlated with root diameter, as predicted by the RES, SRL then mathematically scales negatively with RTD. However, for the region of the nonlinear curve where RTD slowly decreases with root diameter (Fig. 2c), SRL could be positively related to RTD. This is because with increasing root diameter, the negative effect of root diameter could counteract the positive effect of RTD on SRL. In contrast, for the region of the nonlinear curve with fast decrease of RTD (Fig. 2c), SRL may show no relationship with RTD. This is because with increasing root diameter, the negative effect of root diameter on SRL could be offset by the positive effect of RTD on SRL. Together, Introduction L. 81: Throughout the manuscript, the term 'coupled relationships' refers to a negative correlation between root diameter and tissue density, and a positive one between root N and diameter. This is confusing because coupled can also mean opposite trait correlations, especially in the context of a RES. It may be more clear to simply speak of positively, negatively or un-correlated traits? Also, were there any studies included that reported trait "studies reporting no relationship between RTD and root diameter" and "studies reporting negative relationships between RTD with root diameter" (Lines 231-237 in the Discussion section, Lines 726-728 in the Figure legends, and Lines 765-767 in the caption of Fig. 4).
"Data source for studies on woody species: I: studies reporting correlated trait relationships, i.e., negative RTD-root diameter and positive RN-root diameter correlations, both not supporting the root economics spectrum (RES); II: studies reporting un-correlated trait relationships." (Lines 15-16 for the revised footnote (d) of Supplementary Table 1).
Further, a few studies in our dataset reported trait relationships partially in line with the RES, e.g., a positive RTD-RD correlation in Xu (2011) and positive RTD-RD and RN-RD correlations in Valverde- Barrantes et al. (2015). To account for these studies in our analysis of the data source effect, we have revised the Methods and the Supporting information as follows: 5 "There were a few studies reporting above trait relationships only partially supporting the RES (e.g., both positive RTD-root diameter and RN-root diameter relationships, see Supplementary Table 1 for details) on woody species with low RTD. To test whether these few   studies could influence the nonlinear root trait relationships for studies reporting the correlated trait relationships, we tested the trait relationships both with and without these studies using the anova method in R. The influence of these studies was not significant on the nonlinear trait relationships (F=0.93, p=0.54 for the F=0.54,p=0.93 for the RN-root diameter relationships). Therefore, we added these studies to the studies reporting correlated trait relationships." (Lines 412-420 in the Methods section). "There were a few studies for woody species under category I * which partially supported the RES, i.e., reporting positive RTD-RD relationship in Xu (2011) and positive RTD-RD and RN-RD relationships in Valverde-Barrantes et al. (2015)." (Lines 17-18 for the revised footnote (d) of Supplementary L. 105 -111: I agree that trait relationships in the RES needs to take mycorrhiza into account, but how these relationships are affected by EcM roots needs more explanation. Both EcM associations and thicker cell walls have been associated with plants growing in poor environments (L. 105 -109), but is there also a direct relationship (Question 1)? Do EcM roots have thicker cell walls (Question 2)? EcM and AM trees occur in environments that vary in soil fertility, so do EcM trees on richer soils still have denser tissue due to thick cell walls than AM trees on comparably poor soil (Question 3)? Based on this, EcM roots are hypothesized to be denser and lower in N% than AM roots (L. 107 -108), for I assume a given diameter (Question 4)? I am curious about how the effect of fungal mantle on the diameter on EcM roots influences these different trait relationships between EcM and AM roots (Question 5)? If AM and EcM roots of the same diameter are being compared, the EcM roots would mostly include the fungal mantle (as most studies do not correct for this, but see 6 Withington et al. 2006 Ecol Monogr). As this is enriched in N compared to the roots (L. 196) EcM roots may have a higher N% than AM roots? Have the authors considered this effect(Question 6)? In order to better understand the hypotheses, the role of diameter needs to be clarified here. RESPONSE: Thanks for these useful insights! Regarding Questions 1 and 2, we are not aware of any studies directly testing the relationship between EcM status and root cell wall thickness in poor environments. However, there is indeed some indirect evidence. For example, many studies have shown that EcM species usually have high RTD (Chen et al. 2013;Kramer-Walter et al. 2016;Comas et al. 2009). A recent study also showed that roots of some EcM species had higher root tissue density than roots of AM species for a given root diameter (Valverde-Barrantes et al. 2018). EcM hyphal mantle usually has much lower tissue density than root tissues and shows a very weak correlation with root tissue density (r=0.14, p=0.73 for the correlation of the mantle proportion in root cross sectional area with root tissue density; data from Withington et al. 2006). Therefore, the EcM hyphal mantle cannot explain the higher RTD of EcM species than AM species (for Questions 4 and 5). Instead, it is more likely that thickening and/or intensely lignifying root cell walls cause higher RTD for EcM than for AM species. In the revised manuscript, we have added some new discussion to address this point: "As the mantle hyphae usually have low tissue density and have little correlation with RTD 16 , it is expected that for a given root diameter, higher RTD and lower RN in EM species than in AM species could result from thicker and/or more intensely lignified root cell walls in EM species 15,33, 14,15,27,28,29 ." (Line 103-106).

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As for Question 6, we did not consider the influence of the hyphal mantle on RN in EM species for two reasons. First, Compared with AM species, very few studies have explored the relationships between mycorrhizal fungal traits and RN in EcM species. Therefore, we are not convinced about the hypothesized relationship between root diameter and RN in EcM species.
Second, the main aim in this section was to discuss the possibility of differences in root trait relationships between AM and EcM species based on differences in growth habitat (i.e., EcM species usually grow in harsher environments relative to AM species). Therefore, we prefer to discuss the possible role of the hyphal mantle in influencing RN in the Discussion section.
Here, we have provided detailed argumentation for the hyphal mantle effect on RN, and we also refer to the indirect evidence of Withington et al. (2006): L. 125: I agree this work would offer a relevant contribution to the debate on root traits, but it does not settle it -there are still many open questions; for instance, the RES is supposedly driven by the tradeoff between SRL and root lifespan which is not tested in this work, and the differences between woody and non-woody trait relationships are also far from clear. RESPONSE: We agree that the word 'settle' was not appropriate and indeed a bit over-ambitious to cover these questions. We have replaced 'settle' with 'reconcile' (Line 133).
We did not cover the tradeoff between SRL and root lifespan in our study (r=-0.40, p=0.015; figure not shown and very similar to other studies, e.g., Fig. 2a in Weemstra et al. (2016) and Extended Data Fig. 4c in Ma et al. (2018)). To integrate SRL into our argument of the nonlinearity of root trait relationships, we have added a new paragraph: 8 "Furthermore, the nonlinear relationship between root diameter and RTD advances our understanding of another key trait underlying the RES, i.e., specific root length (SRL, root length per unit root mass) 11,49 . Theoretically, SRL can be expressed as: SRL = 4/(π × RTD × root diameter × root diameter) 17,50 . If RTD is positively correlated with root diameter, as predicted by the RES, SRL then mathematically scales negatively with RTD. However, for the region of the nonlinear curve where RTD slowly decreases with root diameter (Fig. 2c), SRL could be positively related to RTD. This is because with increasing root diameter, the negative effect of root diameter could counteract the positive effect of RTD on SRL. In contrast, for the region of the nonlinear curve with fast decrease of RTD (Fig. 2c), SRL may show no relationship with RTD. This is because with increasing root diameter, the negative effect of root diameter on SRL could be offset by the positive effect of RTD on SRL. Together, the nonlinear relationship between RTD and root diameter could explain an overall weak correlation of SRL with RTD, and also with RN (Supplementary Fig. 7a,b) 11,14,15,51 . The weak correlation between SRL and RN could also underlie a weak correlation between root diameter and RN given the wide demonstration of a strong correlation between root diameter and SRL 12,14,15,52 (also see Supplementary Fig.7c). Together, these results illustrate how nonlinear root trait relationships can explain why SRL does not conform to the RTD-related plant economics spectrum in woody species 15,51 ." (Lines 271-288).
As for the difference between woody and non-woody species, we discuss their differences in the Discussion section from the viewpoint of the nonlinearity of root trait relationships.
However, we acknowledge that our previous argument was not very clear and could easily lead to confusing. We have therefore substantially reworked this section: "Compared with absorptive roots of woody species at a given diameter, non-woody species are reported to have about 30% less mycorrhizal colonization 12 . In favorable (e.g., moist and/or fertile) soils, absorptive roots of non-woody species may be less dense 33 (Supplementary Fig. 6) with more root hairs 55 and/or root branching 56 than woody species; this would allow non-woody species to more actively take up nutrients balancing the lower nutrient acquisition through mycorrhizal associations. In contrast, in poor (e.g., dry and/or infertile) soils, roots of woody species may not be much denser than non-woody species because higher RTD would reduce mycorrhizal colonization 18, 57 while woody species are in higher need of mycorrhizal association than non-woody species 12 . However, in poor soils, roots of non-woody species may be denser with thinner stele vessels 33, 58 relative to woody species as relative lower need of mycorrhizal colonization for non-woody species 12 especially for those species with finer absorptive roots 56 . These could explain why roots of non-woody 9 species show greater variation of RTD and RN than woody species at a given root diameter.
This could potentially explain why we did not find nonlinear root trait relationships in non-woody species." (Lines 307-321).
Furthermore, we have also considered the reviewer's comments on the diameter cut-off of 2 mm as a potential factor explaining the weak nonlinearity of root trait relationships in non-woody species compared to woody species: "Another reason for lack of nonlinear trait relationship in non-woody species may be that roots < 2mm in diameter include some non-absorptive roots 59 which typically have larger proportion of stele than absorptive roots 2 , and as such, confound root trait relationships observed for non-woody species. However, absorptive roots of woody species in previous studies (Supplementary Table 1) are sampled based on root branching order which can track the absorptive roots more precisely than the diameter-based method 1 . We therefore recommend for future studies to select absorptive roots based on branching order 1 rather than on root diameter." (Lines 321-328).

Methods
L. 280 -281 : At least for some species (e.g., grasses), still a large proportion of the roots < 2 mm may not be absorptive but transporting and it is considered a questionable threshold "Another reason for lack of nonlinear trait relationship in non-woody species may be that roots < 2mm in diameter include some non-absorptive roots 59 which typically have larger proportion of stele than absorptive roots 2 , and as such, confound root trait relationships observed for non-woody species. However, absorptive roots of woody species in previous studies (Supplementary Table 1) are sampled based on root branching order which can track the absorptive roots more precisely than the diameter-based method 1 . We therefore recommend for future studies to select absorptive roots based on branching order 1 rather than on root diameter." (Lines 321-328).
L. 297 : 'this' should be 'these' RESPONSE: We now mention that these analyses were conducted for woody species only: 11 "In woody species, we also tested whether the data source…" (Line 408).
Studies that reported trait correlations partially in line with the RES were included as follows: "There were a few studies reporting above trait relationships only partially supporting the RES (e.g., both positive RTD-root diameter and RN-root diameter relationships, see Supplementary Table 1 Table 1).

Results
There are many results, figures and tables, and different treatments, so a clear overview is needed. To this end, the Results section needs better structuring. RESPONSE: We have restructured the Results section to make it read more smoothly. First, following the reviewer's suggestion, we give a brief overview of the main results at the end of the Introduction.
"Using a global root trait dataset, we first tested the allometric relationships between root cortex and the stele for both woody and non-woody species. Then, we tested whether the allometric relationship could explain the nonlinearity of relationships of root diameter with RTD and RN for woody and non-woody species and for different mycorrhizal types. Finally, we evaluated how allometry-based nonlinearity may explain the conflicting data on root trait relationships observed in woody species and could reconcile the above debate." (Lines 133-139).
Second, we have moved the paragraph on the role of phylogeny (i.e., Fig. 3 and Table 1, Lines 186-191) towards the end, after the results on growth forms (i.e., woody and non-woody) (Lines 167-185).
Here are some suggestions:

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-The phylogenetic part (L. 143 -146) needs to be presented elsewhere (later), because it breaks the flow of reading the other -more related -results, and is not directly related to the hypotheses (the figure numbers are also not in the right order). RESPONSE: Thanks. We have moved the paragraph on the role of phylogeny (i.e., Fig. 3 and Table 1, Lines 186-191) towards the end, after the results on growth forms (i.e., woody and non-woody) (Lines 167-185).
-I prefer to see Fig  "When accounting for effects of plant phylogeny for all species using phylogenetic independent contrasts (i.e., PICs), RTD was negatively and RN was positively correlated with root diameter (Supplementary Fig. 1a, b)." (Lines 159-162). 13 We have also added 'negatively' and 'positive' in this sentence to indicate the direction of the trait relationships: "Across all species, RTD scaled negatively and nonlinearly with root diameter (R 2 =0.16, p<0.001), and a rather weak positive and nonlinear relationship was found between RN and root diameter (R 2 =0.002, p<0.001) (Fig. 2a "For woody species, the nonlinear relationships…" (Line 192).
"…The results of linear mixed model for woody species showed that…" .
"…in different sets of studies on woody species..." (Line 765). Tables: I prefer   -Use more discernible colors for at least ERM and EM.

Figures and
-Perhaps add a legend on the size of the circles that represent absorptive root diameter (e.g. 3 size classes with the corresponding root diameter value) to get a general idea of the values RESPONSE: We have revised the figure according to these suggestions. In the revised Fig. 3, the left panel looks slightly smaller as the species pool included was smaller than in the right panel.

Discussion
The authors claim their 1st hypothesis was supported by their analyses (L. 179 -180), but this was only the case for the woody species, i.e. 60 % of all species, and even among them, relationships between root N% and diameter were rather weak (L. 148, R2 = 0.002). The results are still very interesting and the nonlinearity of these traits is an important advance in understanding root trait (co-)variation, but the results not/little in line with the hypotheses deserve more attention. Moreover, some of the explanations regarding the results need to be clarified or better supported. As the paper places its results in the context of a RES, relationships with other traits could be more extensively discussed, and it has interesting results to do so. RESPONSE: In the revised version, we have replaced "support" by "partially support".
"Our global analysis of key root traits partially supports our first hypothesis…" (Line 208).
For sections not/little in line with the hypotheses, we have reworked the text to more accurately reflect our findings. For example, we have thoroughly revised our discussion of the relationship between RN and root diameter in EcM species which did not support our hypothesis (Lines 220-228).
Further, we have clarified the Results by using clearer language and more straightforward argumentation. For example, we have clarified the explanation for the weaker nonlinear root trait relationship in non-woody species than in woody species, which was confusing in our initial submission (Lines 307-321).
We have also added more discussion on the implications of nonlinear root trait relationship for another key root trait, i.e., SRL (Lines 271-288).
Please, see below for our more detailed responses to these comments. L. 184: use 'exist' rather than 'existed'; unless this was actually tested? RESPONSE: Done (Line 211).
L. 186 -187: I don't understand this sentence 'Thus, .... relationships'. I don't see what it refers to, and how it relates to growth habit: does growth habit refer to growth form (woody vs non-woody), or something else? The authors should rephrase or explain this. RESPONSE: We apologize for being unclear. Growth habitat ('habit' was not the correct word) refers to the relatively harsher environments for EM and ERM species compared with AM species. Specifically, we initially expected that the nonlinear root trait relationships (e.g. RD-RTD) could be weaker in EM species compared with AM species because EM species usually grow in harsher environments than AM species. For a given root diameter, RTD could be increased in harsher environments by thickening and/or lignifying root cell walls, which could cause a deviation of the RTD from the predicted value based on the allometric relationship (i.e., Fig. 1a in the main text). Therefore, the harsher environments for EM species could result in a weakened RD-RTD relationship for EM roots compared with AM roots. However, our results showed similar RD-RTD relationships between EM and AM species, which did not support these expectations. Therefore, our results suggest that the harsh environments where EM species grow may not exert a strong influence on cell wall thickening as to significantly affect the RD-RTD relationship for EM roots. We have revised this sentence as follows by using 'harsh environments' instead of "habitat": "This suggests that harsh environments may not necessarily exert a strong influence on cell wall thickening for EM and ERM roots in woody species." (Lines 212-214).
L. 194 -197: Please, rephrase or clarify the explanation as to why thick roots of EcM species have lower N% than thin roots (as opposed to Am/ErM species). First, the authors state it is due to '... a lower proportion of roots covered by the hyphal mantle ...'. I guess they mean that thicker EcM roots have lower cover than thin EcM roots, and therefore lower N%? The second explanation refers to the lower nutrient foraging by EMM with increasing diameter. I assume the authors imply that thicker roots of EM species have less foraging precision, less nutrient uptake, and hence, lower root N (based on the literature referred to)? In any case, this needs more and clear explanation in order to interpret the results. RESPONSE: Yes, the two reasons we used to explain the negative root diameter and RN relationship in EcM species are in line with the reviewer's understanding. To further clarify these points, we have rewritten these sentences and added more detailed information on how lower RN in thicker EcM roots can be explained:  L. 212 -213, and L. 220 -221: The 'greater investment of absorptive roots in dry mass' confuses me because greater mass investment can also imply more root length (i.e. at the root system level). From the following lines, the authors seem to refer to greater mass investments per unit root length? RESPONSE: Yes. At the root system level, greater investment in dry mass of absorptive roots can result from: 1) thinner root diameter, greater total root length and thus less root dry mass per unit root length; 2) thicker root diameter, longer single root length (because of the positive correlation between root length and root diameter, Chen et al. 2013;Kong et al. 2014) but less total root length and thus higher root dry mass per unit root length. In the first case, roots may have a shorter lifespan because of their thinner diameter (see Supplementary Fig. 5 and also Ma et al. 2018 Nature), which contradicts our argument of higher root mass resulting in longer root lifespan according to the cost-benefit theory (see Line 246). Therefore, our argument is indeed based on the individual root level, e.g., a single 1 st order root. We have clarified this as follows: "Theoretically, for an individual absorptive root (e.g., a single 1 st order root), greater investment in dry mass could result in longer root lifespan following the cost-benefit theory 43,44 as applied to aboveground plant organs 19,22 ." (Lines 245-247).
Our argument following the above sentence for model in Supplementary Fig. 4 is based on root dry mass instead of root dry mass per unit root length. This is because our argument of "greater investment of absorptive roots in dry mass could result in longer root lifespan (Lines 245-246)" is based on the positive dry mass-lifespan relationship of plant organs (Marbà et al. 2007) as well as the well-recognized cost-benefit theory (Eissenstat et al. 2000). We have clarified this point as follows: "If we consider a plant root a cylinder formed by concentrically arranged tissues, dry mass of a single 1 st order root must be a function of the total diameter as root volume increases exponentially with diameter ( Supplementary Fig. 4)." (Lines 247-249).

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L. 210 -232: The part about other root trait tradeoffs w.r.t. the RES can be further completed: L. 220 -221: The authors state that root diameter rather than RTD influences root mass and hence lifespan. Just to verify, this claim is derived from the fact that root mass still increases with root diameter (Fig. S4), even though RTD decreases with diameter (which would decrease root mass)? As SRL is considered a key trait in the RES, I suggest that the authors discuss the consequences of the nonlinearity of root traits for SRL. The RES hypothesis assumes diameter and RTD to be positively correlated to each other and negatively to SRL.
But if they are not positively related -as shown here -it may explain why RTD is often found uncorrelated to SRL, and why neither SRL and diameter consistently correlate to root N% (Weemstra et al. 2016 New Phyt). RESPONSE: Yes, our claim that "a predominant role of root diameter rather than RTD in determining the root dry mass (Lines 255-256)" is indeed based on theoretical and empirical data (see Supplementary Fig. 4a,b). According to the allometric relationship between root cortex and stele, RTD decreases with increasing root diameter. Therefore, with increasing root diameter, root dry mass can be positively affected by root diameter but negatively affected by RTD. To determine which of the two is more important in determining root mass, we made a model as shown in Supplementary Fig. 4a. This model shows that the negative effect of RTD on root mass can be offset by the positive effect of root diameter on root mass, thus causing root mass to still increase with root diameter. The Supplementary Fig. 4b is an indirect evidence supporting our model in Supplementary Fig. 4a by using data of root diameter and PRS (i.e., proportion of root cross-sectional area occupied by the steles).
To further validate our ideas on the positive relationship between root diameter and root dry mass, we have used a dataset of the single 1 st order root mass (see the Source Data file in ths submission) and root diameter of 96 woody species from our previous study (Kong et al. 2014). The high goodness-of-fit of the cubic regression between the two traits directly and strongly supports the dominant role of root diameter in determining root mass ( Supplementary   Fig. 4c).

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We very much appreciate the comments on our results in relation to SRL. In the revised version, we have added a new paragraph articulating the consequences of the nonlinearity of root traits for SRL: "Furthermore, the nonlinear relationship between root diameter and RTD advances our understanding of another key trait underlying the RES, i.e., specific root length (SRL, root length per unit root mass) 11,49 . Theoretically, SRL can be expressed as: SRL = 4/(π × RTD × root diameter × root diameter) 17,50 . If RTD is positively correlated with root diameter, as predicted by the RES, SRL then mathematically scales negatively with RTD. However, for the region of the nonlinear curve where RTD slowly decreases with root diameter (Fig. 2c), SRL could be positively related to RTD. This is because with increasing root diameter, the negative effect of root diameter could counteract the positive effect of RTD on SRL. In contrast, for the region of the nonlinear curve with fast decrease of RTD (Fig. 2c), SRL may show no relationship with RTD. This is because with increasing root diameter, the negative effect of root diameter on SRL could be offset by the positive effect of RTD on SRL. Together, the nonlinear relationship between RTD and root diameter could explain an overall weak correlation of SRL with RTD, and also with RN ( Supplementary Fig. 7a,b) 11,14,15,51 . The weak correlation between SRL and RN could also underlie a weak correlation between root diameter and RN given the wide demonstration of a strong correlation between root diameter and SRL 12,14,15,52 (also see Supplementary Fig.7c). Together, these results illustrate how nonlinear root trait relationships can explain why SRL does not conform to the RTD-related plant economics spectrum in woody species 15,51 ." (Lines 271-288).
L. 223 -225: Nutrient foraging is suggested to increase with diameter due to a larger cortex relative to stele size. Yet thicker roots are expected to grow slower, reducing their competitive ability to resources from a rich soil patch compared thin-rooted species (Eissenstat 1992, J Plant Nutr;Comas et al. 2014, Frontiers Plant Sci). This other side of the nutrient uptake abilities of thicker roots should be mentioned too.

RESPONSE:
We agree with the reviewer and have added this aspect of nutrient forging for thick roots: "The higher nutrient foraging activity with increasing root diameter may have evolved to compensate for inefficient proliferation of thicker AM roots in resource rich patches 42,46,47,48 ." RESPONSE: We acknowledge that we did not compare the mycorrhizal colonization of absorptive roots between woody and non-woody species. In the revised version, we have revised this sentence by referring to a recent meta-analysis by Ma et al. (2018), which shows on average 30% less mycorrhizal colonization for non-woody species than for woody species (Lines 307-309). However, we agree with the reviewer that mycorrhizal colonization is also an important nutrient foraging strategy in non-woody species. Recent studies show that mycorrhizal colonization in non-woody species is related to some root traits, e.g., negatively correlated with root branching intensity and positively correlated with root diameter (see Li et al. 2017). Therefore, the lower mycorrhizal colonization in non-woody species could be associated with higher root branching intensity for nutrient foraging, especially for non-woody species with finer roots. In the revised manuscript, we have restructured the argumentation concerning the mycorrhizal association in non-woody species. Specifically, we introduce two cases (i.e., rich vs. poor soils) that could contribute to the greater variation in RTD and hence weak or lack of nonlinear root trait relationships in non-woody species: "Compared with absorptive roots of woody species at a given diameter, non-woody species are reported to have about 30% less mycorrhizal colonization 12 . In favorable (e.g., moist and/or fertile) soils, absorptive roots of non-woody species may be less dense 33 (Supplementary Fig . 6) with more root hairs 55 and/or root branching 56 than woody species; this would allow non-woody species to more actively take up nutrients balancing the lower nutrient acquisition through mycorrhizal associations. In contrast, in poor (e.g., dry and/or infertile) soils, roots of woody species may not be much denser than non-woody species because higher RTD would reduce mycorrhizal colonization 18, 57 while woody species are in higher need of mycorrhizal association than non-woody species 12 . However, in poor soils, roots of non-woody species may be denser with thinner stele vessels 33, 58 relative to woody species as relative lower need of mycorrhizal colonization for non-woody species 12 especially for those species with finer absorptive roots 56 . These could explain why roots of non-woody species show greater variation of RTD and RN than woody species at a given root diameter.
This could potentially explain why we did not find nonlinear root trait relationships in non-woody species." (Lines 307-321).
In line with the reviewer's suggestion, we have added another possible explanation for the patterns observed in non-woody species (i.e., the root diameter cut-off point of 2 mm): "Another reason for lack of nonlinear trait relationship in non-woody species may be that roots < 2mm in diameter include some non-absorptive roots 59 which typically have larger proportion of stele than absorptive roots 2 , and as such, confound root trait relationships observed for non-woody species. However, absorptive roots of woody species in previous studies (Supplementary Table 1) are sampled based on root branching order which can track the absorptive roots more precisely than the diameter-based method 1 . We therefore recommend for future studies to select absorptive roots based on branching order 1 rather than on root diameter." (Lines 321-328).
-If root hairs and root branching have a large impact on the balance between cortex and stele, I suspect it would cause variation both between and within woody and non-woody species.
And does this imply that the cortex : stele ratio would be lower for non-woody than woody species (that presumably depend less on mycorrhiza, but still need to transport their resources)? RESPONSE: We have revised our argumentation for root hairs and root branching with regarding to different mycorrhizal colonization between woody and non-woody species (see the following example): "Compared with absorptive roots of woody species at a given diameter, non-woody species are reported to have about 30% less mycorrhizal colonization 12 . In favorable (e.g., moist and/or fertile) soils, absorptive roots of non-woody species may be less dense 33 (Supplementary Fig . 6) with more root hairs 55 and/or root branching 56 than woody species; this would allow non-woody species to more actively take up nutrients balancing the lower nutrient acquisition through mycorrhizal associations" (Lines 307-312).
Different from the expectation of the reviewer, we find that non-woody species have a lower stele:root diameter ratio than woody species (2.4 vs. 2.8, p<0.01). This suggests that non-woody species have a higher proportion of cortex than woody species. The higher proportion of cortex in non-woody could be related to other functions (e.g., more root hairs RESPONSE: The reviewer is correct that these trait relationships only partially support the RES for EM species. Therefore, we have revised this sentence as follows: "…, except for EM trees showing partial support of the RES" (Line 336).

Reviewer 2
In their manuscript "Non-linearity of root trait relationships and the root economics spectrum" the authors compiled a database of root diameter (RD), root tissue density (RTD) and root nitrogen concentration (RN) of absorptive roots of approximately 800 plant species. With this global fine root data they intended to resolve the so far mixed reports on the hypothesized tradeoff between resource acquisition and conservation traits according to the root economics spectrum (RES) and if this differs for woody or non-woody species. In contrast to the 27 predictions of the RES the analysis revealed a non-linear negative relationship between RD and RTD. The authors convincingly argue that this relationship stemmed from the allometric relationship between stele and cortical root tissue which they substantiate with trait information on stele radius (SR) and thickness of tissue outside the stele (tToS) on 158 woody species and 13 non-woody species. In addition, the authors test this non-linear relationship along an evolutionary context and suggest a phylogenetic component in it for woody species with weaker phylogenetic conservatism of RTD than RD. Moreover, their results suggest that ectomycorrhizal woody species show stronger nonlinear relationships than arbuscular mycorrhizal species.
The paper is very interesting and the research questions are timely and very much discussed in the community of root ecologists, functional ecologists and disciplines related to biodiversity research. I also believe the topic of the root economic spectrum and its potential dependence on additional factors is of interest to a wider scientific community. However, this perception is of course stained by personal interests. The study is well composed and the paper very well written. The database is remarkable and data analysis is relatively straight forward and appears to be appropriate although the description could be more detailed in places.
RESPONSE: We greatly appreciate these positive comments, and we are glad to hear that the reviewer thinks our study is very interesting.
My major concern lies with the originality of the results provided in this study. Most of the ideas, hypotheses and results given in this paper have been shown elsewhere already.
Admittedly, this was on more limited data or plant groups or geographic regions but still the major outcomes of the paper are not really new. Even the very concept itself -nonlinearity of the allometric relationship -was published before by the authors (Kong et al. 2017). Results on the phylogenetic relationship of RTD, RD and RN were recently published by Ma and colleagues for roughly 400 species (Ma et al. 2018) and the relationship of phylogenetic structured root traits and mycorrhizal colonization in Valverde- Barrantes (2016Barrantes ( , 2017 . All these papers are correctly cited in the manuscript so I do not want to express here that the authors are not aware of this fact. Also, I am convinced that the more thorough dataset presented here fully warrants a new paper -I just think some of the wording could be better adjusted to the available knowledge. RESPONSE: Thanks for these thoughtful comments. Our study builds on previous work (including our own), but the current manuscript is novel for the following reasons: 1) In Kong et al. (2017) we just presented a hypothesis, i.e., 'the nutrient absorption-transportation balance', to explain the allometric relationship between root cortex and stele which was first demonstrated in our previous study on absorptive roots (Kong et al. 2014). However, our current manuscript uses a unique, very large global dataset to demonstrate the nonlinearity of key root trait relationships based on allometric relationships ( Fig. 1). Furthermore, in our current study, we show that the nonlinearity can explain conflicting results among recent studies on root trait relationships and could potentially reconcile the emerging debates on the RES. This is a big step forward in our global understanding of the ecology, physiology and evolution of absorptive roots.
2) Moreover, our current study addresses several other important points that were neither considered in our previous work nor in other studies. First, differences in the allometric relationships between woody and non-woody species. Our current study shows that both woody and non-woody species follow allometric relationships between root cortex and stele, while the allometries are different (Fig. 1a) and have different consequences for the nonlinearity of root trait relationships for woody and non-woody species (Fig. 2). Second, differences in the nonlinearity of root trait relationships among different mycorrhizal types.
We show that the root diameter vs. RTD relationship is overall similar among AM, EM and ERM species, while the root diameter vs. RN relationship is dramatically different between AM and EM species (see Lines 220-228). Third, the role of phylogeny. Based on the phylogenetic analyses, we demonstrate a phylogenetic component in the nonlinear root trait relationships (Fig. 3, Supplementary Fig. 1 3) The relationships between root diameter and RTD and RN were explored by Ma et al. (2018). However, Ma et al. (2018) only referred to the positive relationship between root diameter and RTD and, importantly, did not considering these relationships within the context of nonlinearity. Furthermore, the allometry of root cortex and stele, an important factor contributing to nonlinearity, was not covered in Ma et al. (2018). climate, phylogeny, growth form and mycorrhizal type on root traits and the relationships between root and leaf traits, but this study did not explore root trait relationships.
The paper is based on a valuable database of root traits for more than 800 species. I am aware that the majority of this data is compiled from FRED yet substantial effort was put into additional data from other published literature. To my knowledge this database is not made publicly available or at least this is not stated in the manuscript or elsewhere in the accompanying material. I consider this fact a major flaw and would strongly encourage the authors to provide the compiled data e.g. to FRED for future use and ability of researchers to reproduce the work given the detail provided in this study.
RESPONSE: Indeed, a large part of the dataset used in our study can be found in FRED. For those studies not included in FRED, we provide detailed information on how to access the data (see Supplementary Table 1 2) Two other datasets also not available in FRED: Xu (2011) (see Table 2-1, Fig. 3-2, Fig. 3-8 and Fig. 5-2) and Jia 2011 (see Table 2.1, Fig. 3.1, Fig. 3.5 and Fig. 3.10); a MSc (thesis in Chinese with figures and tables also shown in English). These trait data are not available in FRED possibly because they will be used for the researches' own studies. However, these two studies can be openly accessed at

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Line 73: I think reference 9 does not support the statement given here RESPONSE: We have replaced this reference with a suitable reference (Ryser et al. 1996 Functional Ecology) (Line 73). "…, we used the mean trait value across these studies as most of the multiple measurements for a species came from the same climatic zones (Supplementary Table 1)." (Lines 351-353).
Moreover, in the revised manuscript, we also account for the climate effect on trait variation by using climatic zone as a fixed actor in the linear mixed model analysis, and the climate effect is not significant: "…we tested the effects of the fixed factors (i.e., root diameter, data source, root sampling, climatic zone, and the study nested within the data source) and the interactions of root diameter with the data source and with root sampling, respectively." (Lines 421-423).
Line 218 ff: You refer to a model presented in the supplement. If I get this correctly your line of argument here is a model shows RD is important for root dry mass. From this you conclude that RD is more important than RTD without further mentioning additional tests and also that root mass and lifespan are positively correlated without reference. So you conclude that there is a relationship between lifespan and RD which had already been shown -so why the model? I do not get the point here I am afraid. RESPONSE: We apologize for being unclear and the confusion this may have caused. Yes, our model ( Supplementary Fig. 4a, see below) is used to demonstrate the dominant role of RD in root dry mass. Actually, Supplementary Fig. 4b provides indirect evidence for testing the role of RD in root dry mass, for which we use data of RD and PRS (i.e., proportion of root cross-sectional area occupied by the stele) presented in Supplementary Table 1. However, as the reviewer points out, we did not provide "additional tests", i.e., tests using root mass and root diameter data. To address this issue, in the revised version, we do provide such a direct test, i.e., Supplementary Fig. 4c where we use the data of 1 st order root mass (see the Source Data file in this submission) and root diameter of the 96 woody species from one of our previous studies (Kong et al. 2014). The new Supplementary Fig. 4c shows that root mass indeed increases with increasing root diameter, further supporting our argument for the dominant role of root diameter in determining root mass. Supplementary Fig. 4 33 Therefore, this model is important as it establishes a connection between root diameter and root dry mass. Given that plant organ mass is usually a proxy for lifespan (see the following paragraph), our model as such can provide a possible explanation for why root diameter is positively related to root lifespan (i.e., via the effect of root diameter on root dry mass).
We acknowledge that we lack direct evidence of a positive correlation between root mass and root lifespan. However, we have some indirect evidence for this argument. First, roots with greater mass but shorter lifespan will be naturally selected against according to the cost-benefit theory (Eissenstat et al. 2000). Second, plant lifespan has been demonstrated to be in a 1/4 power relationship with plant mass across wide range of species 'from tiniest phototrophs to the largest trees' (Marbà et al. 2007). Therefore, although empirical tests are urgently needed, it is reasonable to assume a positive relationship between root mass and root lifespan in absorptive roots. To clarify this point, we have revised this section by referring to the 'cost-benefit theory' and added references (i.e., Marbà et al. 2007, Eissenstat and Yanai 1997, and Eissenstat et al. 2000 as indirect evidence for the positive relationship between the mass and lifespan of plant organs: "Theoretically, for an individual absorptive root (e.g., a single 1 st order root), greater investment in dry mass could result in longer root lifespan following the cost-benefit theory 43, 44 as applied to aboveground plant organs 19,22 ." (Lines 245-247).
Line 224: I thought root foraging capacity is higher with lower RD not increasing with RD? RESPONSE: Our argument here is based on the single root level rather than the root system level (see also our responses to comment of Reviewer 1 to L. 212-213 and L. 220-221 in our initial submission). At the root system level, lower RD may be associated with a greater total root length and hence a greater root foraging capacity. However, at the individual root level (e.g., a single 1 st order root), thicker RD in AM species is usually associated with greater mycorrhizal colonization Kong et al. 2014) and a higher RN because of a lower stele and cell wall proportion (Fig. 1b, Supplementary Fig. 7), both of which indicate higher foraging activity. We have added this clarification to the Results: "Theoretically, for an individual absorptive root (e.g., a single 1 st order root)…" (Line 245).
In the revised manuscript, we also acknowledge some disadvantage of thicker absorptive roots, i.e., the lower root proliferation in resource rich patches, as also pointed out by the Reviewer 1: "The higher nutrient foraging activity with increasing root diameter may have evolved to compensate for inefficient proliferation of thicker AM roots in resource rich patches 42,46,47,48 ." (Lines 263-264).
Line 226: The background argument of higher RN in thicker roots is already in the intro and repeated here, yet you make it sound like a new argument. Perhaps cut this down in the into than? RESPONSE: We prefer to keep this argument here because it follows closely from the argument for "the relative independence between changes in cortical and stele tissues in roots" (Lines 267-268), which indicates much greater variation in the cortex than in the stele with increasing root diameter across species (see Fig. 1a and Valverde-Barrantes et al, 2016).
Line 229: which has been shown before. Sounds here like you show this for the first time.

RESPONSE:
We have changed this sentence as follows: "Together, except for the negative relationship between root diameter and RN in EM species, the nonlinear trait relationships as revealed here suggest that root trait relationships do not necessarily align with the RES hypothesis." (Lines 265-267).
Line 334: give more details on the linear mixed effect models RESPONSE: We have added more details on the linear mixed model by explicitly specifying the fixed and interaction effects: "Then, using a linear mixed model, we tested the effects of the fixed factors (i.e., root diameter, data source, root sampling, climatic zone, and the study nested within the data source) and the interactions of root diameter with the data source and with root sampling, respectively. Climatic zones were classified as tropical, subtropical, temperate, boreal, and mediterranean. Study (see Supplementary Table 1) was not considered as a random factor because they were not classified randomly but assigned to one of the data sources according 35 to their trait relationships. Root sampling referred to studies collecting the 1 st order roots and studies collecting roots up to the 3 rd order, respectively." (see Lines 420-428).
Line 546: you say you use mean values per species for a trait (line 281) and that you have anatomical traits for 13 non-woody species. So why are there more than 13 points for non-woody species in these graphs? Do I miss something here? RESPONSE: In Fig. 1a,b,c the number of species is much greater than 13 (see also Supplementary Table 2). However, there are only 13 species for which both root anatomy and RN have been reported; therefore, we did not show the PRS-RN relationship in Fig. 1d. To avoid any potential confusion, we have added "for this relationship" after "because the sample size". (Line 385). Reviewer #2 (Remarks to the Author): Thanks for this thorough revision of the manuscript. I find the answers to the reviewer requests well supported and overall convincing. I am also content with the revision of the text itself which is now much clearer in reasoning and more consistent in wording and line of argument in most cases. There is still one new part where the SRL is discussed which I find particularly hard to follow. I very much like the new paragraph discussing the absence of a non-linear relationship in non-woody species.
Just a few very minor comments: L 143: I am aware that tToS was defined in the introduction but it is just not a very intuitive abbreviation. I think it would help to define it again in each sections (intro, results, discussion) at first mentioning.
L191: refer to table 1 again at the end of this sentence. You state that: If RTD is positively correlated with root diameter, as predicted by the RES, SRL then mathematically scales negatively with RTD.
I think it is clear till that point. Now you start taking apart the SRL to RTD relationship for the two different regions of the nonlinear relationship. You argue: However, for the region of the nonlinear curve where RTD slowly decreases with root diameter (Fig.  2c), SRL could be positively related to RTD.
To explain this last sentence you state: This is because with increasing root diameter, the negative effect of root diameter (on SRL) could counteract the positive effect of RTD on SRL.
This is not really an explanation for me. You just state different relationships without background as to why the one or the other would be more likely related to biological background. Overall this makes a somewhat confusing and partly circular argument.
Sorry -I only now have the supplementary with the paper and see that figure 7 there does support this argument much better. However, as the figure is in the supplement, I think it is important for this discussion to be understandable without the supporting figure, so please try to make the logic of your argument more explicit and easier to follow.
L 291: Shouldn't it be RTD decreases and RN increases unless I misunderstand the sentence here. Perhaps better put this in two sentences to make it clearer.
L310: Does figure 6 show only non-woody species? Even if so that would not give a comparison to the woody ones directly. Of course you indicate the studies to the figure but I think it is too much to ask the reader checking what species types were in those studies. So please indicate in the figure legend.

REVIEWERS' COMMENTS:
Report from Reviewer #1  15,17,30, We acknowledge that there are few studies reporting direct evidence for the changes of cell walls with soil fertility. Instead, the references we cited here provide some indirect evidence.
We have rephrased the sentence to emphasize this: " ... *may* lead to thicker and/or more intensely lignified root cell walls". Specifically, the cited references show that: 1) Variation of stele cell walls is positively related to RTD: Wahl and Ryser (2000) [Reference 53, r=0.52, p=0.02] 2) RTD is higher in less fertile soils: • L. 309-311: It is suggested that non-woody species change root hairs and branching to acquire resources rather than investing in mycorrhiza compared to woody species. However, woody species also show lower colonization rates on more fertile soils, so I am not sure if this explains the differences between the growth forms (Question 1). Moreover, the manuscript does not clarify how this is related to a reduction in the RTD of non-woody species (L. 310); perhaps a part of the rebuttal that explains the link between cortex size (and thus RTD) and hairs and branching could be briefly incorporated(Question 2)?
RESPONSE: Thanks for these insight comments and suggestions! As for Question 1, we note from Ma's study (see the above figure) which showed a constant lower mycorrhizal colonization in non-woody than in woody species across a wide range root diameters (slopes are equal; intercept is lower in non-woody than in woody species). As roots in this study were sampled from a wide variety of ecosystems (subtropical forests, temperate forests, grasslands) across a large gradient of soil fertility (Wang et al. 2018), it is likely that the lower mycorrhizal colonization in non-woody compared to woody species holds true for both fertile and infertile soils.
As for Question 2, we have taken the reviewer's advice and added part of the text from our first rebuttal letter regarding the cortex size: As for the second point, Ma et al. (2018) showed constantly lower mycorrhizal colonization in non-woody than in woody species across a wide range of root diameters (see figure above).
As roots in this study were sampled from a wide variety of ecosystems (subtropical forests, temperate forests, grasslands) across a large gradient of soil fertility (Wang et al. 2018), it is likely that the lower mycorrhizal colonization in non-woody compared to woody species holds true for both fertile and infertile soils.
• L. 318-320: 'These ...' speculations/findings...? Also, roots of non-woody species do not necessarily have greater variation in RTD and root N than woody species for a given diameter based on Fig. 2 and R 2 =0.026) ( Fig. 4; Supplementary Data 2)."  We have also added both terms to the legend of Figure 4 (Lines 683-687). However, for the region of the nonlinear curve where RTD slowly decreases with root diameter (Fig. 2c), SRL could be positively related to RTD.
To explain this last sentence you state: This is because with increasing root diameter, the negative effect of root diameter (on SRL) could counteract the positive effect of RTD on SRL.
This is not really an explanation for me. You just state different relationships without background as to why the one or the other would be more likely related to biological background. Overall this makes a somewhat confusing and partly circular argument.
RESPONSE: Thanks for these thoughts. We are sorry for any confusion we may have caused.
We will clarify these points below: (1) Our reasoning is based on the nonlinear and negative relationship between RTD and root diameter. Under this nonlinear relationship, the two components of SRL (i.e., RTD and root diameter, Ryser 2006) have opposing effects on SRL according to the following mathematical relationship (Ryser 2006;Holdaway et al. 2011). There are two reasons for why negative effects of root diameter could counteract potential positive effects of RTD for the region of the nonlinear curve with a slow decrease of RTD with increasing root diameter. 1) A smaller positive effect of RTD relative to the larger negative effect of root diameter on SRL with increasing root diameter.
2) The negative effect of root diameter on SRL would be magnified as root diameter is in second power whereas RTD is in first power in the denominator of the above formula.

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(2) Our argumentation for the relationship of SRL with other root traits is based on the nonlinear relationship between RTD and root diameter. As we demonstrate, such nonlinear relationship is derived from the allometric relationship between root stele and cortex tissues.
Our argumentation in this section concentrates on the region of the nonlinear curve with a slow decrease of RTD with increasing root diameter (see Fig. 2c). Here, the slow decrease of RTD can be attributed to the slow decrease of the proportion of root cross-sectional area accounted for by root stele (Fig. 1b). We argue that less stele tissue or more cortex tissue with increasing root diameter might result from stronger dependence of nutrient foraging on mycorrhizal fungi for thicker absorptive roots Liu et al. 2015;Eissenstat et al. 2015). We mention this biological explanation in Lines 270-279.
(3) Our argumentation is not circular. In this section of our manuscript, we try to understand the relationships of SRL with RTD and RN, as shown in previous studies, according to two trait relationships, i.e., our finding of the nonlinear relationship between RTD and root diameter, and the mathematical relationship of SRL with RTD and root diameter: 4/(π × RTD × root diameter 2 ). These two trait relationships are essentially different from each other because 1) the formula of SRL = 4/(π × RTD × root diameter 2 ) is derived from the definition of SRL and RTD and the assumption that roots are cylindrically shaped; 2) the nonlinear relationship between RTD and root diameter is derived from the allometric relationship between root stele and cortex tissues, and is based on some biophysical mechanisms (see Kong et al. 2017). This means that the formula of SRL = 4/(π × RTD × root diameter 2 ) holds regardless of the nonlinear relationship between RTD and root diameter, and vice versa.
Therefore, there should be no circular argument for our discussion in this section.
We have simplified the explanation in the main text: "However, for the region of the nonlinear curve where RTD slowly decreases with root diameter (Fig. 2c), the negative effect of root diameter on SRL could counteract a potential positive effect of RTD on SRL, which, in turn, would lead to a positive relationship of SRL with RTD. In contrast, for the region of the nonlinear curve with fast decrease of RTD (Fig.   2c), SRL may show no relationship with RTD. This is because with increasing root diameter, the negative effect of root diameter on SRL could be offset by the potential positive effect of RTD on SRL." (Lines 286-293) Sorry -I only now have the supplementary with the paper and see that figure 7 there does support this argument much better. However, as the figure is in the supplement, I think it is important for this discussion to be understandable without the supporting figure, so please try to make the logic of your argument more explicit and easier to follow. RESPONSE: We initially explored whether the poor relationships of SRL with RTD and RN (i.e., Supplementary Figure 7a,b), as shown in previous studies, could be explained by our finding of allometry-based nonlinearity combined with the formula of SRL = 4/(π × RTD × root diameter 2 ). Here, we try to make this logic more explicit and easier to follow: 1) We speculate that RTD would show different relationships with SRL for the region of the nonlinear curve with slow decrease and for the region of the curve with fast decrease of RTD with increasing root diameter. This could lead to a weak relationship between RTD and SRL across the whole range of nonlinear curve as shown in Supplementary Figure 7a.
2) RTD is usually negatively correlated with RN because higher RTD will be accompanied with greater investment of nitrogen in the cell wall fraction (see Supplementary Figure 6), thus leading to lower root activity and RN. Given the weak relationship between RTD and SRL, SRL would also be weakly correlated with RN (Supplementary Figure 7b).
3) The weak correlation between SRL and RN could also contribute to the weak correlation between root diameter and RN (see Fig. 2 in the main text) because SRL is strongly coupled with root diameter (Supplementary Figure 7b).
We have revised the relevant sentences as follows: "Together, the nonlinear relationship between RTD and root diameter could explain an overall weak correlation of SRL with RTD, and also with RN ( Supplementary Fig.   7a,b) 11,14,15,49 across the whole region of the nonlinear curve. Moreover, the weak correlation between SRL and RN and the strong coupling of SRL with root diameter 12,14,15,50 (also see Supplementary Fig.7c) could also explain the relative weak correlation between root diameter and RN (Fig. 2)." (Lines 293-300).
L 291: Shouldn't it be RTD decreases and RN increases unless I misunderstand the sentence here. Perhaps better put this in two sentences to make it clearer.
RESPONSE: Our apologies, the reviewer is correct. We have corrected this mistake.