## Bulk properties of the model orchard tree roots

Table 2 summarizes the bulk architectural properties of the root models, which include the total volume, total surface area, total length, buried depth and the number of generations, among others. The model orchard trees have a similar trunk diameter of ~ 7–8 cm (Table 2). The total root volume ranges from 1,983 to 4,673 cm3, with the total root volume increasing with an increase in the trunk diameter because the volume of the main trunk (tap root) comprises a large portion of the total volume (e.g., 30–50% of the root models). The total surface area and length are 4,994–9,807 cm2 and 1,112–3,100 cm, respectively. The buried depths range from 38–65 cm; Lovell 2 and 3, Marianna 1 have shallow depths of ~ 35–40 cm while Lovell 4, Marianna 3 and 4 have depths as high as ~ 60 cm. The three year old trees bifurcate 7–10 times on average. As shown by these bulk properties, there are no clear distinct differences among the Lovell, Myrobalan, and Marianna species.

Figure 5 shows the cumulative volume and surface area versus the distance from the trunk center at the ground surface (or trunk top). A hemisphere with the center at the top of the trunk can be imagined; and, the volume and surface area of the root enclosed by the hemisphere are calculated as a function of the radius of the hemisphere. As shown in Fig. 5, the cumulative root surface area and volume increase as the hemisphere radius is increased. The maximum distance from the trunk top to the root tip is less than approximately 120 cm.

The slope of the relationship between root volume or surface area with its distance from the trunk top is related to the fractal dimension of the root architecture, as follows:

Root Volume or Area = F∙Rτ, (1)

where R is the distance from the trunk top, F is a constant, and τ is the fractal dimension. Both the volume and the area show a bilinear trend with the distance from the trunk in log-log scale plots, and the region with the hemisphere radius from 10 cm to 30 cm, which is close to the trunk with active bifurcation, is chosen to determine the fractal dimension. It is found that the fractal dimension ranges 1.1–1.5 for the volume, and 1.6–2.2 for the surface area (Fig. 5). The fractal dimensions for the root volume are consistent with the previously reported values; *e.g.*, 1.17–1.66 for the coarse root systems of *Grewia flava, Strychnos cocculoides, Strychnos spinosa, Vangueria infausta* and 1.85 for the 6-week old tomato root system (Eshel, 1998; Oppelt et al. 2000). It is noted that this study uses the hemisphere-counting method while those previous studies used the box-counting method for the fractal dimension analysis. To the authors’ knowledge, there is no reported data on the fractal dimension for the surface area.

## Angles of branches at bifurcation nodes

Root grow and extend through bifurcations. When bifurcation occurs at a node, two child roots grow from one parent branch. From the perspective of the bifurcation node, two types of relative angles with respect to the parent root can describe the direction of the child roots in three dimensions: the branching angle and the relative azimuthal angle between the parent root and one of the child roots. Therefore, at every bifurcation, two branching angles and two relative azimuthal angles are defined for two child roots.

Figure 6a presents the branching angle values determined from the model roots, and Fig. 6b shows the average values with associated standard deviations. When the parent root is vertical to the ground, a branching angle of 0° indicates the child root that grows down vertically and parallel with the parent root. A branching angle between 0° and 90° means that the child root grows diagonally down, and a branching angle between 90° and 180° corresponds to a child root that grows upwards toward the ground surface. The main lateral roots at a generation of 2 have the greatest branching angle, with a mean value close to 70°, implying that the child roots grow close to a horizontal direction (Fig. 6b). The branching angle decreases with an increase in generation number after generation 2, and the lateral roots at the last four generations have average branching angles between 10–30° (Fig. 6b). This indicates that the lateral roots at later generations bifurcate almost in parallel with minimal branching angles between the parent and child roots. This trend is consistent with previous studies with *A. lenticularis, A. nilotica, A. procera, D. sissoo, P. dulce, S. grandiflora, C. fistula*, and *S. cumini* (e.g., Chaturvedi and Das 2003), in which the primary roots have the greater branching angles than the secondary roots, and hence spreading of the primary roots is greater than that of the secondary roots.

The histograms of branching angles of entire root systems can be captured with lognormal distributions, as shown in Fig. 7. The mean branching angle for the trunk roots ranges from 35° to 66° for the main lateral roots. Most importantly, the lognormal distributions fitted to the data show that the main lateral branching angles are generally significantly larger than for the remaining laterals, resulting in radial extension away from the trunk axis.

Figure 8 shows the histogram of relative azimuthal angles for all child branches at all bifurcation points. The distribution reveals that the relative azimuthal angles roughly show a uniform distribution from 0° to 360°. This implies that the cardinal directions of bifurcated child roots are close to random regardless of root types (i.e., trunk, main lateral, and remaining lateral roots).

## Branch length of roots

Figure 9 depicts the branch lengths and their mean values of all the model roots as a function of generation number. As mentioned before, the branch lengths are determined as the distance between the bifurcation nodes from the root skeletons. The branch lengths of the trunk roots are the shortest, mostly less than 10 cm (Fig. 9a). This implies frequent bifurcations over the course of tap root growth; on average, tap roots bifurcate every 1.6 cm. From the second generation (Generation 2), the branch length significantly increases, showing a range of average values between 10–30 cm (Fig. 9b).

Figure 10 shows the histograms of branch length values from all the root models. The results can be reasonably fitted with exponential distributions with the mean values differing depending on the root classification. The roots having a generation number greater than 2 have the greatest mean value of 21 cm (Fig. 10c) while the main lateral and trunk roots have values of 17.6 and 1.6 cm (Figs. 10a and 10b), respectively. This is consistent with the root growth patterns of the orchard trees, in that the roots grow longer before bifurcation for effective lateral coverage as they grow further away from the trunk. The 3D models shown in Fig. 1 visually corroborates this pattern.

## Distribution of root diameters: local thickness analysis

The root diameters are examined based on the diameters of the maximum inscribed spheres (MIS) that fit within the 3D models. This MIS diameter is also called the local thickness (LT). Figure 11 shows the cumulative distributions of root diameters for the model roots. Note that the median value of the root diameter, *LT50*, corresponds to the 50th percentile of root volume fraction. The median root diameters range from 14.5 to 33.5 mm. Lovell 4 shows the maximum *LT50* of 33.5 mm while Lovell 2, Myrobalan 3, and Marianna 3 have the minimum *LT50* of 14.5 mm. Table 3 summarizes local thicknesses of roots. Particularly, the ratio of *LT60* to *LT10* can be used as an indicator to root diameter variations, how broadly the root diameters are spread. Similarly, in a discipline of geotechnical engineering and soil mechanics, the coefficient of uniformity, Cu, for the grain size distribution is defined as the ratio of *D60* to *D10*, and it has long been used as an engineering indicator to how widely the grain size varies for a given soil. The analysis reveals that the coefficient of uniformity (Cu) of the model roots, defined as *LT60*/*LT10* in this study, ranges from 2.7–5.6.

Table 3

Summary of local thickness

*Symbol* | *Lovell 1* | *Lovell2* | *Lovell3* | *Lovell4* | *Marianna 1* | *Marianna 2* | *Marianna 3* | *Myrobalan 1* | *Myrobalan 2* | *Myrobalan 3* |

LT10 | 5.5 | 5.5 | 6.5 | 7.5 | 6.5 | 8.5 | 5.5 | 7.5 | 7.5 | 3.5 |

LT30 | 12.5 | 9.5 | 11.5 | 17.5 | 11.5 | 17.5 | 9.5 | 11.5 | 11.5 | 8.5 |

LT50 | 21.5 | 14.5 | 19.5 | 33.5 | 19.5 | 25.5 | 14.5 | 16.5 | 16.5 | 14.5 |

LT60 | 30.5 | 20.5 | 27.5 | 39.5 | 27.5 | 33.5 | 22.5 | 20.5 | 20.5 | 17.5 |

LT90 | 79.5 | 85.5 | 68.5 | 78.5 | 69.5 | 57.5 | 70.5 | 38.5 | 38.5 | 32.5 |

Cu *a* | 5.55 | 3.73 | 4.23 | 5.27 | 4.23 | 3.94 | 4.09 | 2.73 | 2.73 | 5 |

Note: *a* The coefficient of uniformity in root thickness Cu is defined as LT60/LT10, which represents the variability in thickness. The Cu value of 1 indicates that the root thickness is uniform. Greater Cu indicates greater variation of root thickness. |

The local thickness gradient is estimated using the local thickness values at two bifurcation nodes and the distance between two bifurcation nodes, *i.e.*, ∇LT = (LT1 – LT2)/L2. Figure 12a shows the local thickness gradient ∇LT with respect to the branch length L2. The local thickness gradient decreases with an increase in the branch length, and all ∇LT values are positive. The result also shows that majority of ∇LT values are smaller than 0.5. Figure 13 shows the histograms of local thickness gradients for trunk, main lateral, and remaining lateral roots of entire root models, and they are described with the exponential distributions. The mean values of ∇LT for the trunk and main lateral roots are 0.35 and 0.32, respectively. In contrast, the mean ∇LT is 0.1 for remaining lateral roots. This result demonstrates that the root diameter decays at a significantly faster rate near the trunk. It is worth noting that the trunk roots and main lateral roots (generations 1 and 2) have roughly similar mean values of ∇LT, while their mean branch lengths have large differences (1.6 cm versus 17.6 cm, as denoted in Fig. 10).

The root cross-sectional area ratio at a bifurcation node is defined as the square of local thickness ((LT22 + LT32)/ LT12). Such area ratio allows examination on the conservation of cross-sectional areas of a root across bifurcations. The area ratio is equivalent to the reciprocal of the proportionality index (*p*). Figure 12b shows the area ratio with respect to the mean branch length ((L2 + L3)/2). Herein, the area ratio is estimated with the main lateral roots and the lateral roots while the trunk root less than 3 cm long and the end branch are excluded. In most instances, the area ratio is close to 1, with most values ranging between 0.9 and 1.2, which implies the conservation of area across a bifurcation. This agrees with the pipe stem theory, which contends that this hydraulic architecture naturally evolves to preserve cross-sectional areas for efficient water and nutrient transport (Oppelt et al. 2001; Van Noordwijk et al. 1994). In addition to natural variability, there can be deviations in the hydraulic architecture from the external diameter owing to the thickness of non-conducting material encasing the root (Danjon et al. 1999, 2013; Coutts 1983). There are some cases where the area conservation is less than 1, and this low area ratio is more frequent as the branch length increases. This is simply because this area ratio estimation measures the thickness at the bifurcation nodes, therefore, the long branch length between the bifurcation nodes renders underestimation of the area ratio particularly. That is, the next bifurcation nodes (LT2 and LT3) are located too much far away from the current bifurcation node (LT1).