Materials and Methods

2.1 Classical growth equations for the rising growth trend

The Gompertz and logistic equations are two important types of sigmoidal trajectories. In order to better describe the unimodal trajectory, here, we discretized these two equations. For the Gompertz model, we first introduced a time parameter T to re-describe growth rate, whereT refer to the formation time of unit tissue. Although still considered conceptual, T is not biologically arbitrary, but describes the time during which tissue development is controlled by genes and physiological activity, independent of size. Assumingf (M ) is the total biomass of new tissues during timeT , then the expression term of growth rate isf (M )/T in this period of time. In mathematics,T can also be infinitely small so thatf (M )/T → dM /dt. So the introduction ofT has a broader significance. Applying this concept and the Bertalanffy paradigm, we not only can obtain the discrete Gompertz equation, but also extend it further. The discrete logistic equation can be directly derived from the population model based on the intrinsic growth rate and reproductive generations. See supplementary information for the detailed derivation steps of all equations. The resulting discrete growth expressions can be written as:
(1a)
(2)
Eq. 1a is the extension of the discrete Gompertz equation, and can be simplified to
(1b)
Where f (M )/T represent the growth rate; (ignoreo ) and parameter o ensures that the growth equation can move along the transverse axis, reflecting the modular strategy of tree growth. The vertex of Eq. 1b is (, ).
Since T , gr and b are relatively stable, M max is mainly determined bycM /mr . By definition,cM ∝ resource uptake, 1/mrand 1/gr . Obviously, Eqs. 1a and 2 belong to the Bertalanffy family.
The effects of parameterscM , b ,T ×mr /gr,M max and λ on growth curves are shown in Figs. 1a and b. Note that the parameter cMreflects the average levels of resource uptake and respiration consumption. The change ofT ×mr /gr mainly depends on mr , because mris more sensitive to the environment (Van Iersel 2003). Mathematically, unimodal curves connected in series can be divided into types αand β, as shown in Fig. 1c. Type α indicates that unimodal curves in series have the same starting point. In type β , the starting point of the new curve can be anywhere on the old curve, reflecting modular growth. Biologically, the difference between αand β types lies in the involvement of the old tissues in the new growth pattern. Considering Eq. 1 as an example, continuous changes in parameters (cM andT ×gr /m r ) may result in a mixed trajectory, as shown in Fig. 1d. We referred to this type as type γ.
We speculated that changes in the relationship between functional traits and tree size may affect cascading growth, with both positive and negative effects (PE and NE) on growth. The effects of PE and NE on Eqs. 1 and 2 are related to changes in functional traits such as module and overall performance, as shown in Table. 2. On the module scale, we considered both leaf and stem economics (Table. 2), which are related to photosynthesis, hydraulic transport, and respiration consumption. These traits are important for tree ontogeny (Westoby 1998; Weiher et al. 1999; Poorter et al. 2008; Héraul et al., 2011). On a more comprehensive scale, we considered the number of trait modules relative to size, where some morphological traits that are related to the total amount of photosynthesis and respiration deserve special attention. The product of average crown width/(DBH+crown width) (abbreviated as CW/(D+CW)) and crown height/tree height (i.e., crown ratio) (abbreviated as CH/H) can be considered a proxy of canopy/size. Note that the size in this indicator is not biomass but volume. Higher canopy/size means a resource intake strategy, then increase cM andM max. The attenuation of light within the canopy will cause a decrease in light utilization after the canopy closes, so the uptake rate of trees to resources is not directly proportional to their canopy.
Sapwood is the living, outermost portion of a woody stem or branch, and heartwood is the dead, inner wood, which often comprises the majority of a stem’s cross-section. Sapwood, not heartwood, serves as a sink tissue that consumes sugars for daily metabolism through respiration (Lehnebach et al., 2016; Bamber et al., 1976). Because heartwood is not active, themr of the trunk could decrease with a decreasing sapwood/heartwood ratio. Given that the tree trunk accounts for most of the tree biomass, we calculated the sapwood area to heartwood area ratio to assess the change of tree mr . Obviously, the change of mr can affect maintenance respiration (= mr × biomass) from the scaling relationship with biomass. In fact, sapwood maintenance respiration may also be a significant cause of growth decline with increasing tree age (Tatuo and Shidei, 1967; Ryan and Yoder, 1997; Sillett et al., 2010). Overall, these two ratios have great effects on plant growth.

2.2 Site description and experimental design

We studied four subalpine primevalAbies fabri forests located in the Gongga Mountain, in the southeastern Tibetan Plateau, and at altitudes of 2,900 m, 3,000 m, 3,300 m, and 3,600 m (treeline). The mean annual temperature was 4 °C, and the mean annual precipitation was 1,938 mm, with roughly 50% of the annual total rainfall occurring from June to September in 2015. The characteristics of these survey plots are described in Wang et al (2017).
Abies fabri trees have large stature and long life history, and we expected to observe significant biomass changes on the century scale. First, we estimated the ideal and average growth trajectories (with respect to size) of this species by the DBH sequence of all sampled trees and biomass equations. These results can be used to test H1 and H2. Note that the ideal growth trajectory is composed of the maximum growth increment of different trees along the size gradient. Then, we focused on changes in some important functional traits with tree size. This can be achieved by analyzing the functional traits of different trees. Some variable functional traits may be the key to cascading growth. Finally, we tried to quantify the effects of these variable functional traits on unimodal growth to test H3. Sampled functional traits is current, so they are only valid for recent growth dynamics. Thus, some individual trees that follow the unimodal trajectory only in recent growth would be selected, all located at an altitude of 3,100 m. We expect that the height and length of these unimodal trajectories (related to parameters cM ,mr , λ and M max) are determined by some current functional traits (Tab. 1).

2.3 Data collection, processing and analysis

In September 2015, we measured the tree ring sequence and DBH (> 10cm) of each tree in four large plots along elevation gradients. For details of the method, see Wang et al. (2017). In June 2019, we sampled and measured the relevant functional traits of trees belonging to different DBH classes. DBH classes are in the 5 cm interval. These randomly sampled trees covered more than half of the sample areas surveyed in 2015. Meanwhile, the functional traits of the selected trees were sampled separately. For leaf and stem economics (listed in Tab. 1), we sampled three times at the south and north sides of trees, and at different heights (e.g., 5m, 10m,15m). The number of leaves sampled each time ranged between 20 and 60. Some leaves were used for element analysis. Leaf N concentrations were analyzed with a Vario MAX CN element analyzer (Elementar, Hanau, Germany). The vanadium molybdate yellow colorimetric method was used to measure Leaf P, and flame photometry was used to measure Leaf K. We used foliar dry matter content to locate different trees on a resource use axis rather than specific leaf area (SLA) (Wilson et al., 1999) due to the large vertical span of the fir canopy and sampling difficulty. Details of trait measurement are described in Wilson et al. (1999) for foliar dry matter content. For stem economics, trunk xylem density was calculated as the ratio of mass of samples dried for 72 h at 105℃ after the removal of bark, to their fresh volume as calculated by the displacement of liquid volume (water). Trunk bark thickness was measured by vernier caliper. Trunk wood moisture was determined by the fresh mass and dry mass of the wood, and was expressed as fresh mass/dry mass-1. We determined the sapwood width using diameter increment borers. Sapwood and heartwood areas are equal to π×(DBH/2)2-π×(DBH/2-sapwood width)2and π×(DBH/2-sapwood width)2, respectively. Other morphological traits were obtained by direct forestry survey.
We used the actual tree ring sequence to determine the aboveground biomass dynamic of individual trees. To reduce error, the minimum time interval for describing growth dynamic was set to 10 years. The increase in DBH for each decade can be derived from the current DBH and tree-ring data. Based on previous results (Zhou et al., 2013), we established the aboveground biomass equation of individual trees dependent on DBH, which allows us to obtain the growth trajectory of individual trees (see supplementary information). From these results, we can determine the best and average DBH or biomass increment within different DBH classes (> the maximum diameter increments) to reveal the growth dynamic along along the DBH or size gradient. The emphasis of maximum increment is on species upper-quantile growth change (95% percentile) (Wright et al., 2010), reflecting a relatively ideal growth state.
For Eq. 1b, f (M )/T can represent the annual average growth rate over a decade. Thus, the 10-year aboveground biomass increment can be approximately equal to 10f (M )/T .
(3)
Where b = 0.75. Assuming a constant ratio of aboveground to underground biomass (Shu et al., 2019), individual growth equations can also be used directly to describe the change of aboveground biomass. We used Eqs. 3 and 2 and their extended forms (see supplementary information) to fit the growth dynamics of aboveground biomass and DBH to directly test the validity of classical growth equations. If these equations are valid for the selected trees, we can get the length and height of two type of unimodal curves (Figs. 1a and b), which are closely related to parameter λ , M amax10mr /gr andcM . According to Tab. 1, we can directly test the effect of selected variable traits on the formation of unimodal curve.