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The median diameter of a longitudinal section of the stem may be used to determine the stem volume. However, to calculate stem volume, many measurements of diameter at different heights along the stem are required. Therefore, this approach is not generally applied because time-consuming and expensive. Here, we propose a novel, more rapid method to obtain median diameter using the area of the stem profile. A total of 218 height/diameter classes from more than 5000 spruce trees (^{2 }= 0.9985). Statistical analysis revealed that the height of the median diameter on the stem profile was 0.3

Various methods are used to estimate the volume of both standing and felled trees. Of the many methods applied to estimate the bole volume of felled trees, Smalian’s, Huber’s and Newton’s formulae are usually adopted. All three methods provide good results for a frustum of a paraboloid and a cylinder (

Nonetheless, some equations use sections other than the cross-sectional area at the midpoint, or the cross-sectional area at the lower and upper end of the bole (

If the median diameter value of the stem profile is known, the stem volume may be computed using the Mathiesen’s formula (

where _{i}). The area of each section, _{i} (_{i}) is based on _{i} represents the diameter at the middle of the sections

This study aimed to develop a more rapid method for assessing the median diameter of the stem profile of Norway spruce trees (

A database was established in 2012 by compiling the stem diameters of 5403 Norway spruce trees from Romania, grouped into 218 diameter-height classes (Tab. S1 in Supplementary material). Original measurements were performed by the FRMPI (Forest Research and Management Planning Institute of Romania) before 1950 in 200 plots, located in the Romanian Carpathians. The database consists of Norway spruce trees with breast height diameter (_{DM} - _{DMrel} -

The median diameter of the stem profile (

where_{DM1} and _{DM2} are the areas with values under and above _{1} and _{2} are the corresponding values for diameters of _{DM1} and _{DM2}, respectively. The parameter _{DM} can be estimated using

where _{DM1} and _{DM2} are the distances between the lower end of the trunk and the point where _{1} and _{2} occur. Consequently, _{DMrel} may be derived as follows (

From the previous work (

A correlation analysis was applied to test the association between the median diameter at breast height (_{DM}, and _{DMrel}. Similarly, the association of tree height (_{DM}, and _{DMrel} was also tested. Moreover,

The coefficient of variation within each breast height diameter class was computed for height, median diameter of the stem profile, and absolute and relative height of the median diameter, in order to assess data homogeneity.

The evolution of the median diameter of the stem profile as a function of breast height diameter for Norway spruce trees in Romania is shown in ^{2 }= 0.9985;

The same values of median diameter grouped by total tree height indicated a large variation for each tree height class (^{2}) was only 0.5523. The best fitted model in this case was a polynomial expression (

(^{2 }= 0.5523;

The relationships of the absolute height of the median diameter with the diameter at breast height (^{2 }= 0.4573; ^{2 }= 0.9876;

The low accuracy of the first model (_{DM}, as depicted in

A positive correlation was detected between the absolute height of the median diameter and tree height (r = 0.9937).

The core of this study is the analysis of the relationship of the relative height of median diameter (_{DMrel}) with _{DMrel} values were mainly distributed along and around a horizontal asymptote (intercept = 0.2961, _{[216]} = 103.0696,

The mean _{DMrel} value computed using all the 218 tree size classes was 0.298695 ± 0.001936 (95% confidence interval), with a low coefficient of variation (4.88%), indicating fairly homogenous samples.

The results revealed that the median diameter of the longitudinal sections of the stem could be determined as approximately 0.3 × H in the case of Norway spruce trees in Romania.

The relationship between the relative height of the median diameter along the stem and total tree height (^{2 }= 0.5344;

The coefficient of variation was computed for each breast height diameter class of each studied variable of the stem profile: tree height, median diameter of the longitudinal section, and absolute and relative height of median diameter.

The classes of diameter at breast height were homogenous for all the studied variables. The coefficient of variation was lower than 30% in most cases, ranging between 13.99% and 25.78% for total height, and between 15.73% and 31.64% for the absolute height of the median diameter on the stem. For comparison, the median diameter values (range: 0.60% to 1.64%) and the relative location of median diameter values (range: 2.04% to 6.74%) were significantly lower. Due to high data homogeneity, arithmetic means of the _{DMrel} mean values was 0.2978. The observed variation of values in all 25 breast height diameter classes was extremely low (coefficient of variation = 1.99%).

The median diameter of the stem profile assessed using

We also calculated the relative deviation of median diameter estimates obtained with the two aforementioned methods using both

_{DM}, and _{DMrel} were best predicted by using both independent variables (

Our results indicated that two aforementioned methods could be used to obtain a rapid assessment of median diameter of the stem profile for Norway spruce trees in Romania, specifically: (i) using

(ii) measuring the stem diameter at 0.3

The first method only involves measuring breast height diameter and a simple calculation for both standing and felled trees. The second method requires the measurement of tree length (considered to be equal to tree height), followed by the direct measurement of the median diameter at 0.3

The practical implications of the proposed methods are relevant, particularly when the relative height of the median diameter is correlated with the distribution of volume in trees. These methods should be applied to other species to validate their use, especially in the case of conifers. The Mathiesen’s formula for volume estimation could be modified and improved by using the proposed models. Our results demonstrated the feasibility of assessing the stand volume using the median diameter of the stem profile. For two species of pines (loblolly and ponderosa pine),

The median diameter of the stem profile could be used to study stand architecture and competition between trees (

Compared to

Resistance of a tree depend on many factors.

Our study developed two methods to assess the median diameter of the stem profile without measuring multiple diameters at different heights along the stem. The median diameter may be obtained without classical computations that require the area of the stem profile. We demonstrated that diameter at 0.3 × height is a good approximation of the median diameter. Accuracy analysis of the proposed models indicates differences < 2% in most cases. Tree distribution based on the median diameter of the stem profile represents a new direction for forest dynamics studies. The ratio between the median diameter of stem profile and the tree height (

We thank Andres Kiviste and Allan Sims (Institute of Forestry and Rural Engineering, Estonian University of Life Science) for providing Mathiesen’s original paper (1925). We thank Paolo Cherubini (WSL Swiss Federal Institute for Forest, Snow and Landscape Research), Tudor Stancioiu, and Alexandru Borz (Faculty of Silviculture and Forest Engineering, Transilvania University of Brasov) for providing valuable comments on the manuscript.

MMV originally formulated the concept and elaborated the database; MMV and CCT developed the methodology; MMV, FD, and CCT analysed the data; MMV formulated the mathematical models; and MMV and FD wrote the manuscript.

Position of the median diameter (_{DM}): height of the median diameter along the stem; (_{i}): area of the

Regression analysis of the median diameter of the stem profile in

Relationship of the absolute height of the median diameter along the stem profile with: (a) breast height diameter; and (b) tree height. Best fitting models are shown by solid lines (

Values of the relative height of median diameter along the stem with respect to: (a) the class of breast height diameter; and (b) the tree height class. The best fitting model is shown by a solid line (_{DMrel} = 0.2961 (

Relative difference (%) of median diameter estimates with: (a) the class of breast height diameter; (b) the tree height class. The depicted model for predicting median diameter values is:

Schematic representation of the stand structure using the median diameter of stem profile as a descriptor. The horizontal projection of the cross-sectional area at the height of the median diameter of trees could be considered an alternative approach to study tree growth, tree resistance to wind, and forest dynamics.

Predictive models of median diameter and its absolute and relative height along the stem obtained by multiple regression analysis.

Best fitted models | ^{2} |
df | F | P |
---|---|---|---|---|

0.999055 | 2, 215 | 113689.80 | < 0.0001 | |

_{DM} = -0.703391 - 0.029121 |
0.998636 | 2, 215 | 78681.77 | < 0.0001 |

_{DMrel} = 0.264829 - 0.000983 |
0.873587 | 2, 215 | 742.88 | < 0.0001 |

Tab. S1 - Classical computation of the median diameter.