1. Introduction

Plant growth is a fundamental ecological process that is affected by physiology, community dynamics, and ecosystem properties (Paine et al., 2012). For a forest ecosystem, the large-tree growth trend determines the carbon budget of forests, especially old forests. Tree growth has been thought to follow a ‘rise-and-fall’ unimodal pattern (i.e., the tree biomass increment per unit time first increases and then decreases with tree size), thus presenting a sigmoidal trajectory of biomass over the lifetime of a tree (e.g., Ryan and Yoder, 1997). Most of the data from chronosequence-based studies supports a decline in biomass accumulation at the community scale (e.g., Acker et al., 2002; Ryan et al., 1997). However, some studies of very old trees suggested an increase in the growth increment with tree size or age (e.g., Sillett et al., 2010; Johnson et al., 2009), which directly contradict the unimodal or sigmoidal growth model. Although there is obvious inconsistency in both aggregate and individual growth trends (Sheil et al., 2017), the root cause of the conflict remains unknown. Overall, technical or/and theoretical limitations have limited a more comprehensive understanding of the growth behavior of large trees (Sheil et al., 2017).
Plant biomass accumulation represents the balance of photosynthetic gains and respiratory losses. Increased total respiratory expenditure is the main cause of productivity decline, which is generally considered to conform to von Bertalanffy paradigm (Von Bertalanffy, 1957; West et al., 2001; Shi et al., 2013). Gompertz and/or logistic equations based on the this theoretical framework (see below) reasonably explain the growth of most small plants (e.g., crops) (e.g., Shi et al., 2013; Karadavut et al., 2008). These results support the rationality of the unimodal pattern, and impels us to consider some significant differences between trees and other small plants. Compared with small plants, large trees have larger-scale physiological and ecological activities, resulting in stronger and more flexible self-regulation ability. For instance, for an average stand, tree height growth, as a game-theory problem (Falster et al., 2003), depends on the presence of the height strategies of other trees. Compared with large trees, the shorter life history of small plants means there are fewer opportunities to be affected by external factors (e.g., disturbances). Thus, small plants are more likely to maintain their inherent growth strategies, thus presenting a growth pattern that is consistent with the theory. More flexible growth strategies, in contrast, may delay achievement of photosynthetic-respiration balance.
Plant functional traits are measurable morphological and physiological attributes that significantly affect whole-plant performance (Enquist et al., 2007). Classical life history theory indicates that at least four independent trait-defined axes (leaf-height-seed-stem) (Baraloto et al. 2010b) can shape the core strategies plants use to acquire and invest resources (Falster et al., 2018). For example, wood density is related to biomass allocation and photosynthetic carbon gain (Santiago et al., 2004) and stem economics and adult stature largely explained interspecific differences in growth strategies for rain forest tree species (Héraul et al., 2011). To maximize the efficiency of using system resources, increased size may shift the benefits and costs of some trait-based trade-offs, resulting in a size-dependent change in the net effect of a particular trait on growth (Falster et al., 2011). A recent meta-analysis supported this idea, showing that trait-growth correlations change with plant size (Gibert et al., 2016). Moreover, functional balance allows an individual to adjust its growth strategy to effectively respond to its immediate environment within the envelope of possibilities defined by allometry (Chen et al., 2013). For example, trees may make substantial plastic adjustment in morphology and anatomy of newly developing leaves, xylem, and fine roots to respond to environment stress (West 2019). If an increase in the investment of valuable traits, plant resource use efficiency may increase as a result. Meanwhile, for trees with obvious changes in size, the relationship between these traits and plant size will change significantly. Due to the indeterminate and modular growth of plants (Weiner 2004), we speculated that the growth trajectory of large trees may include two or more unimodal curves with different scales (i.e., cascading growth). On the whole, the downward trend of the former unimodal trajectory may be obscured by the larger upward trend of the latter trajectory. In essence, cascading growth is result from plant exhibit a range of phenotypes depending upon its environment (acclimation) (i.e., phenotypic plasticity).
The measurement of tree biomass remains a difficult challenge. Researchers have used multiple indicators (e.g., diameter and tree height) or a specific empirical equation to estimate tree biomass. However, the range of a measured indicator limits the applicability of empirical equations (Sileshi 2014). Variation in architecture and form, ontogeny, bark thickness, wood density, damage, and rot all contribute to variation within and among species (Sheil et al., 2017). Thus, it is appropriate to estimate the biomass of trees with fitness and competitive advantage (termed ideal trees) using calibrated equations. These ideal trees grow in relatively ideal environments, so they can invest more resources in growth and approach the ideal growth pattern. The functional traits of these trees must be optimal to promote growth. Obviously, if the growth trajectories of ideal trees are unimodal, then other non-unimodal growth trajectories may represent the incomplete expression of a unimodal pattern. This may be related to the relatively weak influence of functional traits on growth. The ideal growth pattern should be considered separately rather than included in the average results.
We tried to determine whether the observed continuous growth uptrend can be attributed to cascading growth. We hypothesized that: 1) the ideal growth pattern of trees is unimodal (H1); 2) tree growth trajectories may follow unimodal curves with cascade characteristics, and the scale of new curves should increase with tree size (H2), and 3) cascading growth is due to the beneficial change of functional traits relative to tree growth (H3). Based on extended classical growth equations, these hypotheses were specifically tested by analyzing the growth dynamics of different subalpine Abies fabri forests in western China.