Tree structure equations derived from pipe model theory (PMT) are well-suited to estimate biomass allocation in Scots pine ( Pinus sylvestris L.) and Norway spruce ( Picea abies [L.] Karst.). However, age dependence of parameters should be accounted for when applying the equations. Pipe model theory-based (PMT-based) structure equations have been incorporated in many process-based models. However, more data concerning old-growth trees is needed to test the reliability and generality of the structure equations. This study (1) tested the age independence of the PMT-based structure equations and (2) provided general information about the stability of tree structure with age. A total of 162 Scots pine and 163 Norway spruce trees in four age groups were analysed to test the age effect on the parameters of structure equations using a linear mixed model. Biomass of stem, branch and foliage was estimated from destructive measurements, and with other tree dimensions, they were used to present the tree growth patterns. (1) Stem biomass proportion increased with age, while branch and foliage biomass proportion decreased; biomass allocation and most tree variables became steady after maturing. (2) PMT-based structure equations were well-suited to Scots pine and Norway spruce in all age groups; however, age dependence was detected in the parameters of these equations, except for the branch-related equations in Scots pine and stem form coefficient below the crown base in both species. Our study (1) provides information applicable to predictions of growth and biomass allocation in old boreal stands and (2) suggests taking age effect into account when structure equations are implemented in forest growth models.

中文翻译：

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年龄对苏格兰松（Pinus sylvestris L.）和挪威云杉（Picea abies [L.] Karst.）树结构和生物量分配的影响

源自管模型理论 (PMT) 的树结构方程非常适合估算苏格兰松 (Pinus sylvestris L.) 和挪威云杉 ( Picea abies [L.] Karst.) 的生物量分配。然而，在应用方程时应考虑参数的年龄依赖性。基于管道模型理论（基于 PMT）的结构方程已被纳入许多基于过程的模型中。然而，需要更多关于老树的数据来测试结构方程的可靠性和通用性。本研究 (1) 测试了基于 PMT 的结构方程的年龄独立性，(2) 提供了关于树结构随年龄变化的稳定性的一般信息。分析了四个年龄组的 162 棵苏格兰松树和 163 棵挪威云杉树，以使用线性混合模型测试年龄对结构方程参数的影响。茎、枝和叶的生物量是通过破坏性测量估计出来的，而对于其他树木尺寸，它们被用来呈现树木的生长模式。(1)茎生物量比重随年龄增加而增加，而枝叶生物量比重下降；生物量分配和大多数树木变量在成熟后变得稳定。(2) 基于 PMT 的结构方程非常适合所有年龄组的苏格兰松和挪威云杉；然而，在这些方程的参数中检测到年龄依赖性，除了苏格兰松中的分支相关方程和两个物种的冠基以下的茎形状系数。