^{*}

Conceived and designed the experiments: DD GA. Performed the experiments: DD. Analyzed the data: DD GA. Wrote the paper: DD GA.

The authors have declared that no competing interests exist.

Computational modeling of neuronal morphology is a powerful tool for understanding developmental processes and structure-function relationships. We present a multifaceted approach based on stochastic sampling of morphological measures from digital reconstructions of real cells. We examined how dendritic elongation, branching, and taper are controlled by three morphometric determinants: Branch Order, Radius, and Path Distance from the soma. Virtual dendrites were simulated starting from 3,715 neuronal trees reconstructed in 16 different laboratories, including morphological classes as diverse as spinal motoneurons and dentate granule cells. Several emergent morphometrics were used to compare real and virtual trees. Relating model parameters to Branch Order best constrained the number of terminations for most morphological classes, except pyramidal cell apical trees, which were better described by a dependence on Path Distance. In contrast, bifurcation asymmetry was best constrained by Radius for apical, but Path Distance for basal trees. All determinants showed similar performance in capturing total surface area, while surface area asymmetry was best determined by Path Distance. Grouping by other characteristics, such as size, asymmetry, arborizations, or animal species, showed smaller differences than observed between apical and basal, pointing to the biological importance of this separation. Hybrid models using combinations of the determinants confirmed these trends and allowed a detailed characterization of morphological relations. The differential findings between morphological groups suggest different underlying developmental mechanisms. By comparing the effects of several morphometric determinants on the simulation of different neuronal classes, this approach sheds light on possible growth mechanism variations responsible for the observed neuronal diversity.

Neurons in the brain have a variety of complex arbor shapes that help determine both their interconnectivity and functional roles. Molecular biology is beginning to uncover important details on the development of these tree-like structures, but how and why vastly different shapes arise is still largely unknown. We developed a novel set of computer models of branching in which measurements of real nerve cell structures digitally traced from microscopic imaging are resampled to create virtual trees. The different rules that the models use to create the most similar virtual trees to the real data support specific hypotheses regarding development. Surprisingly, the arborizations that differed most in the optimal rules were found on opposite sides of the same type of neuron, namely apical and basal trees of pyramidal cells. The details of the rules suggest that pyramidal cell trees may respond in unique and complex ways to their external environment. By better understanding how these trees are formed in the brain, we can learn more about their normal function and why they are often malformed in neurological diseases.

Dendritic morphology underlies many aspects of nervous system structure and function. Dendrites, along with axons, define the connectivity of the brain

Despite its importance, dendritic branching remains poorly understood

Computational modeling offers a complementary approach to traditional molecular means of uncovering fundamental properties of dendritic branching (e.g.,

While data driven simulations have increased our understanding of dendritic development, they are difficult to compare directly. Different studies often focus on separate structural levels or details, and are rarely based on the same cell classes. Here we expand on previous approaches by testing a suite of three closely related models, both individually and in hybrid combinations. Also, because data driven modeling generally requires quality neuronal reconstructions, they tend to be limited to one or two dendritic tree types. From these studies, it is often difficult to determine how general the results are, and to discern biological insights from data or model peculiarities. With a large digital database of neuromorphological reconstructions now, online (NeuroMorpho.Org), we were able to apply our models to a wide variety of dendritic trees (

Sample cells showing the variety of tree morphologies used as data for this study.

The core of our modeling approach is a recursive branching process as described in

(A) Flowchart showing how virtual trees are created from sampled basic parameters. (B) Depiction of the basic parameters. (C) Depiction of the three fundamental determinants which constrain the sampling of the basic parameters. (D) Morphometrics which are neither fundamental nor basic are emergent to the model and can be employed to compare the real and virtual trees.

The term “fundamental determinants” is meant to describe the parameters which are primary in the model and drive the selection of other values, but should not be taken to imply that they are the only or most crucial developmental factors underlying branching behavior. The comparative approach constrains the choice of fundamental determinants to those compatible with the common mechanics of the model. Nevertheless, the chosen determinants are biologically important and have all been implicated by earlier studies (reviewed in

In earlier efforts (e.g.

Example basic parameter (daughter diameter ratio) distributed irrespective of fundamental parameters (inset), as used in previous studies, and the same parameter binned by Path Distance (main plot: columns and error bars are means and standard deviations, respectively). Both the main graphs and the inset only include daughter ratio values greater than one. The solid line (secondary axis in the main plot) shows the percentage of unitary values in each bin. The dotted line represents the overall percentage of unitary daughter ratios.

The creation of three individual models with the same underlying mechanics also allows the implementation of hybrid variations. This step overcomes some limitations of the simpler models by introducing more freedom, but complicates biological interpretability. Most importantly, the details of how the mix models improve upon (or do not) the individual models provides information on the individual models themselves. We explored two alternatives (detailed in

The comparative application of different but related models to extremely diverse morphological classes enables us to look both within and across cellular/subcellular features for parameter interactions. These interactions may then point to important developmental principles. Four biologically important morphometrics which are emergent to the model are used to compare the real and virtual trees (

The three individual models were evaluated in terms of their ability to produce virtual trees with values of the emergent morphometrics that best matched the corresponding real trees. Strong trends were shown when considering all of the tree classes together (

The upper portion of each panel shows the average relative difference between the means of virtual trees and those of real trees for each fundamental determinant (RAD = Radius, PD = Path Distance, BO = Branch Order). The lower portions show the proportion of times each model is the best (as measured by the distance metric) at determining the emergent morphometrics for the 68 tree groups. The fundamental determinants differ in their relative ability to capture each of the emergent morphometrics. The asterisk (*) signifies p<.05 for all figures as determined by the Mann-Whitney U non-parametric test. All error bars show standard error. (A) Number of bifurcations is best captured by Branch Order. (B) Branch Order is significantly worse than Radius or Path Distance at capturing bifurcation asymmetry. (C) Surface area is captured equally well by all three fundamental determinants. (D) Path Distance best captures surface area asymmetry.

Overall, bifurcation asymmetry was best determined by both Path Distance and Radius (

These trends were generally robust throughout individual tree groups. However, a finer analysis organized by morphological classes revealed additional insights. The tree groups were first divided into apical (n = 18), basal (n = 18), and non-pyramidal (n = 32). The Branch Order model was significantly better than either Radius or Path Distance at determining the number of bifurcations in both basal and non-pyramidal tree types (

(A) Apical trees have their number of bifurcations best captured by Path Distance (RAD = Radius, PD = Path Distance, BO = Branch Order). (B) Basal and (C) non-pyramidal trees have their bifurcation numbers best determined by Branch Order. This may point to different underlying developmental mechanisms between apical and other tree types. (D, E) Sholl-like plots showing bifurcation number as a function of branch order for sample apical (D) and basal (E) groups of cortical pyramidal cells (Markram layer 4, N = 24). Path Distance better captures apical bifurcations while Branch Order better captures basal arbors.

The situation is almost reversed if models are evaluated based on another emergent morphometric, namely bifurcation asymmetry instead of the number of bifurcations (

(A) Apical trees have their bifurcation asymmetry best determined by Radius (RAD = Radius, PD = Path Distance, BO = Branch Order). (B) Basal trees have their bifurcation asymmetry best determined by Path Distance, which wins over the other two models two-thirds of the time. (C) Non-pyramidal trees lie somewhere in the middle, with neither Path Distance nor Radius giving better bifurcation asymmetry results. (D, E) The values of bifurcation asymmetry vary as a function of branch order in representative apical (D) and basal (E) groups (Amaral CA1, N = 23). Path Distance better captures the basal pattern, while the Radius model better captures apical asymmetry.

While

(A) Number of bifurcations is better captured by both Radius and Branch Order in basal than in apical trees. (B) Conversely, bifurcation asymmetry is better captured by both models for apical trees. In either case, non-pyramidal trees tend to lie in between apical and basal trees. (C) The relative ability of the individual models to differentiate apical from basal trees is greater than for other tree divisions. The Performance Ratio is the absolute value of the log of the ratio between the two tree types of the mean differences between real and virtual trees. Number of bifurcations is shown as positive bars (black), bifurcation asymmetry as negative bars (gray). With models based on Branch Order and Radius, the apical-basal divide shows the largest performance ratios for both bifurcation number and asymmetry. The numbers above the Radius columns represent the count of tree groups for the corresponding divisions.

Such a measure also allows the comparison of different criteria to divide neuronal groups besides basal and apical, such as other cellular classifications (e.g. pyramidal and non-pyramidal), developmental stage (young and adult), animal species (rat and others), or median-based metrics (with respect to e.g. size and symmetry). The ability of the different models to differentiate between apical and basal trees is much greater than for other divisions tested (

Tree Type | Lab | Cells | Trees | Bif # | Bif Asym | Surf Area (μm^{2}) | Surf Asym |

CA1 Apical | Amaral | 23 | 30 | 46.20 | 0.60 | 8053 | 0.53 |

Claiborne | 7 | 8 | 44.00 | 0.57 | 36787 | 0.49 | |

Guylas | 18 | 18 | 49.61 | 0.61 | 12553 | 0.56 | |

Larkman | 6 | 7 | 38.43 | 0.59 | 18074 | 0.52 | |

Turner in vivo young | 24 | 25 | 46.16 | 0.59 | 18362 | 0.57 | |

Turner in vitro aged | 15 | 18 | 62.50 | 0.57 | 24268 | 0.55 | |

Turner in vitro young | 10 | 12 | 50.58 | 0.52 | 22053 | 0.54 | |

CA1 Basal | Amaral | 23 | 77 | 7.77 | 0.35 | 1340 | 0.27 |

Claiborne | 7 | 24 | 7.75 | 0.36 | 6927 | 0.23 | |

Guylas | 18 | 62 | 6.85 | 0.35 | 1257 | 0.23 | |

Larkman | 6 | 35 | 6.31 | 0.40 | 2297 | 0.40 | |

Turner in vivo young | 24 | 75 | 10.08 | 0.42 | 3641 | 0.33 | |

Turner in vitro aged | 15 | 48 | 9.52 | 0.41 | 3516 | 0.34 | |

Turner in vitro young | 10 | 33 | 8.06 | 0.39 | 3269 | 0.35 | |

CA3 Apical | Amaral | 24 | 42 | 22.86 | 0.51 | 6504 | 0.46 |

Barrionuevo | 8 | 11 | 24.91 | 0.50 | 9188 | 0.48 | |

Jaffe | 6 | 6 | 26.33 | 0.48 | 26699 | 0.40 | |

Turner | 18 | 23 | 21.13 | 0.50 | 15844 | 0.41 | |

CA3 Basal | Amaral | 24 | 99 | 9.03 | 0.39 | 1714 | 0.26 |

Barrionuevo | 8 | 33 | 7.21 | 0.36 | 2281 | 0.24 | |

Jaffe | 6 | 19 | 7.84 | 0.40 | 5966 | 0.25 | |

Turner | 18 | 61 | 10.46 | 0.38 | 6771 | 0.34 | |

Cortical Pyramidal Apical | Markram layer 2/3 | 37 | 43 | 14.35 | 0.50 | 4094 | 0.48 |

Markram layer 4 | 24 | 21 | 11.43 | 0.50 | 3593 | 0.54 | |

Markram layer 5 | 22 | 23 | 57.43 | 0.60 | 17701 | 0.61 | |

Wearne local young | 20 | 20 | 17.85 | 0.51 | 3381 | 0.47 | |

Wearne local old | 17 | 17 | 18.59 | 0.47 | 5053 | 0.48 | |

Wearne long young | 24 | 24 | 22.88 | 0.49 | 3282 | 0.47 | |

Wearne long old | 19 | 19 | 17.74 | 0.48 | 3495 | 0.48 | |

Cortical Pyramidal Basal | Markram layer 2/3 | 37 | 167 | 3.44 | 0.33 | 887 | 0.29 |

Markram layer 4 | 24 | 114 | 2.77 | 0.30 | 854 | 0.47 | |

Markram layer 5 | 22 | 143 | 3.13 | 0.38 | 942 | 0.39 | |

Wearne local young | 20 | 108 | 3.51 | 0.35 | 716 | 0.29 | |

Wearne local old | 17 | 96 | 3.90 | 0.28 | 722 | 0.28 | |

Wearne long young | 24 | 152 | 3.96 | 0.34 | 839 | 0.32 | |

Wearne long old | 19 | 122 | 3.70 | 0.31 | 751 | 0.31 | |

Dentate | Claiborne | 43 | 73 | 8.89 | 0.38 | 11518 | 0.20 |

Gyrus | Turner in vivo | 19 | 37 | 8.32 | 0.43 | 4304 | 0.33 |

Granule | Turner in vitro | 19 | 38 | 6.92 | 0.43 | 4072 | 0.37 |

Cortical Interneuron | Guylas calbindin | 18 | 69 | 2.78 | 0.30 | 2119 | 0.26 |

Guylas cck | 14 | 61 | 4.15 | 0.30 | 8368 | 0.25 | |

Guylas calretenin | 29 | 83 | 2.64 | 0.26 | 1600 | 0.23 | |

Guylas parvalbumin | 20 | 88 | 2.64 | 0.31 | 4683 | 0.15 | |

Jaffe lacunosum-mol. | 13 | 53 | 3.91 | 0.32 | 3879 | 0.32 | |

Jaffe radiatum | 13 | 50 | 4.40 | 0.43 | 3928 | 0.33 | |

Jaffe other | 17 | 68 | 4.50 | 0.40 | 2485 | 0.33 | |

Markram | 23 | 139 | 2.58 | 0.34 | 784 | 0.30 | |

Turner | 13 | 43 | 4.63 | 0.41 | 2222 | 0.43 | |

Purkinje | Martone | 4 | 5 | 282.20 | 0.50 | 10352 | 0.50 |

Rapp | 3 | 3 | 435.33 | 0.50 | 45679 | 0.54 | |

Spinal Motoneuron | Ascoli p3 | 9 | 59 | 11.69 | 0.46 | 4024 | 0.47 |

Ascoli p11 | 8 | 65 | 9.06 | 0.44 | 1608 | 0.44 | |

Burke | 6 | 69 | 13.77 | 0.47 | 54717 | 0.51 | |

Cameron 1–2 day | 10 | 56 | 3.09 | 0.41 | 2471 | 0.32 | |

Cameron 5–6 day | 12 | 83 | 2.08 | 0.31 | 2652 | 0.28 | |

Cameron 14–15 day | 14 | 47 | 2.81 | 0.39 | 3747 | 0.31 | |

Cameron 19–25 day | 8 | 82 | 2.33 | 0.28 | 1922 | 0.31 | |

Cameron phr 2 week | 5 | 63 | 3.76 | 0.36 | 5791 | 0.32 | |

Cameron phr 1 month | 6 | 66 | 3.36 | 0.33 | 6943 | 0.30 | |

Cameron phr 2 month | 5 | 56 | 6.11 | 0.40 | 11382 | 0.39 | |

Cameron phr 1 year | 6 | 62 | 6.66 | 0.40 | 27434 | 0.40 | |

Fyffe alpha | 8 | 89 | 7.45 | 0.41 | 25796 | 0.41 | |

Fyffe gamma | 4 | 29 | 3.48 | 0.37 | 14513 | 0.24 | |

Retinal Ganglion | Miller small simple | 16 | 60 | 3.07 | 0.34 | 494 | 0.32 |

Miller small complex | 5 | 38 | 7.21 | 0.49 | 2152 | 0.39 | |

Miller medium simple | 15 | 10 | 14.00 | 0.47 | 704 | 0.48 | |

Miller med. complex | 25 | 122 | 9.47 | 0.46 | 1291 | 0.39 | |

Miller large complex | 4 | 14 | 12.64 | 0.44 | 3157 | 0.43 |

The left four columns show the mean emergent morphometric values for each group: number of bifurcations, bifurcation asymmetry, surface area, and surface asymmetry.

After comparing the ability of the “pure” fundamental determinants to control virtual growth and the emergence of various morphometrics in different cell classes, we examined the effect of mixing the influences of Branch Order, Radius, and Path Distance in the hybrid models. The “% Mix” model combines the three fundamental determinants in each of 66 fixed proportions, and samples the basic parameters according to the respective weights. In the “243 Mix” model, every basic parameter can be controlled by a different fundamental determinant. For any tree group and emergent morphometric, the best individual variants of each of these two hybrid models are singled out. Even if all variants were statistically equivalent in their ability to reproduce the morphology of real trees, better quality can be expected because of the sheer number of repetitions (and the selection of the winner). Thus, in order to compare the two hybrids and the best individual models fairly, each of the three approaches was “normalized” to the same number of 243 iterations (with varying random seeds), and the best result was chosen in each case.

The general trend across all 68 cell groups is that the 243 Mix clearly outperforms the best individual model, with the % Mix yielding somewhat intermediate results depending on the emergent morphometric (

(A) The ability of the different model variants to capture the emergent morphometrics. The best individual (BI) and percent mixing (% Mix) were repeated with different random number seeds until they produced 243 virtual tree groups for every real one to match the number produced in the determinant mixing paradigm (243 Mix). The determinant mixing paradigm, where the sampling of each basic parameter could be controlled by a separate fundamental determinant, was significantly better at capturing bifurcation asymmetry and total surface area. Both mixing paradigms were better than the best individual models at capturing surface area asymmetry. (B) Sample real and virtual dendrograms using the determinant mixing paradigm. Scale bars are the same for each real-virtual pair.

The relative weights of the fundamental determinants in the winning combination of the two hybrid models for each emergent morphometric reflects the trends observed when examining the performance of the pure models. Specifically, we compared the fraction of tree groups “won” by each individual determinant with the proportions of the winning % Mix model and the composition of the 243 Mix. Averaging the results over all tree types reveals similar values of the three determinants from the three protocols within any one morphometric property (

The top row compares the percentage of winning best individual models (BI) to the relative contribution of the three fundamental determinants (RAD = Radius, PD = Path Distance, BO = Branch Order) to the winning models in the percent mixing (% Mix) and determinant mixing (243 Mix) paradigms for (A) number of bifurcations, (B) bifurcation asymmetry, (C) surface area, and (D) surface area asymmetry. The bottom row shows how the fundamental determinant contribution to the winning 243 Mix model breaks down by basic parameter (DR = daughter-ratio, PDR = parent-daughter-ratio, TR = taper rate, BPL = branch path length, BIF = bifurcation probability). The overall trend in the determinant mixing paradigm is for a more even distribution of fundamental determinant influence than seen in the best individual and percent mixing paradigms. The basic parameters with fundamental determinant weights close to those seen in the best individual model are likely the strongest drivers in the best individual model selection.

Sampling each basic parameter using a separate fundamental determinant, the 243 Mix model provides an opportunity to gain additional insights into how specific aspects of dendritic structure and development can interact to produce mature morphologies. In particular, it is instructive to analyze how the makeup of the 243 hybrid breaks down for the five basic parameters across the emergent morphometrics throughout all cell types (

When capturing bifurcation asymmetry, Branch Order contributes above average to taper rate and branch path length, Radius to daughter ratio and parent-daughter ratio, and Path Distance to bifurcation probability (

Dendritic development is a complicated process (reviewed in

Most previous modeling attempts varied widely in both their core methodology (i.e. the specifics of the algorithm and the choice of variables) and in the cell classes they attempted to recreate (see

The general results link individual fundamental determinants to the specific emergent morphometric they each best capture, and provide a baseline for comparing particular tree types. The number of bifurcations is best described by Branch Order and worse by Radius. Biologically, the cell may have the ability to “count” branch order locally when determining whether to bifurcate again, possibly detecting the partition of available downstream resources at each bifurcation. The poor performance of Radius suggests that a constant taper rate relating to steady microtubule loss is not a primary mechanism to limit or arrest branching. However, Radius is a better performer than Branch Order with regards to bifurcation asymmetry. Radius may modulate asymmetry by allowing larger branches to bifurcate while their smaller sisters terminate. Interstitial branching, the formation of side branches off of existing branches, constitutes a potential biological underpinning, as it typically produces a larger diameter disparity than terminal branching (the splitting of an extending growth cone). Path Distance can also regulate asymmetry if all branches terminate equidistant from the soma (symmetric trees), or form a distal tuft of bifurcation (asymmetric trees). This may relate to the transport of intracellular messengers or reaction to localized extracellular signals. Since only Path Distance fully succeeds in capturing surface area asymmetry, Radius may be missing vital length or position dependence. Finally, the equal contribution of all fundamental determinants to surface area suggests that this emergent morphometric is not specifically constrained by any individual corresponding biological correlate.

A limitation in regards to the interpretation of results is inherent in the restricted amount of data available in each individual group of cells. This scarcity prevents the practical or statistically meaningful investigation of the branching behavior of all neuronal classes separately. Therefore our analysis concentrated on sub-groupings of the 68 unique datasets. The groups were divided based on a wide variety of criteria, including emergent parameter values, laboratory of origin, animal species and age, brain region, and arbor type (apical, basal, or non-pyramidal). In addition to investigating the relative model performance of many of these divisions by hand, the ability of all of the model variants to capture emergent morphometrics was subjected to cluster analysis. The resulting groups were systematically compared to the above divisions as well as visually inspected for other meaningful classification criteria. Of all the various tree groupings consistent with the available collection of real morphologies, the model performance was only statistically differentiated between apical and basal dendrites (

One important aspect to note is that pyramidal cells, as opposed to many of the other modeled tree types, grow in a very layer specific manner (as seen graphically in

The morphological response of dendrites to NTs and other chemicals is very complex (reviewed in

An alternative or additional mechanism that could underlie the differential performance of various models in the simulation of apical and basal trees involves shifting competition for an intracellular signal or cytoskeletal metabolite. Previous statistical analyses have provided convincing indication that dendritic branching may be homeostatically regulated by global and local competition for limited intracellular resources

As flexibility is added to the models by allowing the different fundamental parameters to contribute to a single virtual tree through model mixing one would expect an improvement in the virtual emergent morphometrics. Both bifurcation asymmetry and surface area were significantly better reproduced by the 243 Mix paradigm than by either the % Mix or individual models (

There are several dimensions in which this work could be expanded. While we are trying to gain developmental insights, digital reconstructions of real cells in publicly available databases are currently limited to adult (or at least relatively mature) neurons

This study raised the possibility that apical and basal dendrites differ from each other due to the histological environment through which they extend, while the morphologies of non-pyramidal cells might be more intrinsically driven. By expanding the suite of fundamental determinants to include planar and radial distance from the soma, this hypothesis could be tested more directly. Such an extension would require 3D embedding of the virtual cells (see e.g.,

As they occur in different parts of the same cells, the striking contrast between apical and basal trees may be costly to control and achieve, and is likely to be relevant from the information processing standpoint. This puts renewed emphasis on the question of what this divide could facilitate in the brain. Due largely to methodological considerations, the relatively thin basal branches are seldom investigated in electrophysiological experiments. Even modeling studies tend to concentrate on different divisions of the apical tree (e.g.

In this study, morphometric parameters that control dendritic branching are measured from groups of real cells and resampled stochastically to create virtual trees of the corresponding class. The real neurons consist of 736 digital reconstructions from 16 different labs. The apical and basal trees of pyramidal cells are treated separately, summing up to a total of 68 individual groups (

Virtual trees in the form of dendrograms are generated with a simple recursive algorithm (

Except for the “unique” case of the initial diameter, the basic parameters extracted from different portions of real trees vary considerably

Aside from the bifurcation probability (a scalar fraction), each bin is then fitted by least square error to the best of three 2-parameter functions: gamma, Gaussian, and uniform. In a previous study

Two types of hybrid models were also tested by “mixing” the fundamental determinants. In the “% Mix” model, each fundamental determinant contributes a percentage of influence over the sampling of the basic parameters. These percentages are varied for each fundamental determinant from 0% to 100% at 10% increments. For example, Branch Order may contribute 10%, Path Length 70%, and Radius the remaining 20%. This sums up to 66 distinct variants of the % Mix model including the “pure” (unmixed) models. For the basic parameters controlling diameter, the probability of sampling a value of one is first computed as the weighted average of the three individual probabilities. For all basic parameters not determined to be one, values are sampled from all three fundamental determinant distributions and averaged together based on their relative weights. In the second mixing method, each basic parameter depends on a different fundamental determinant. For example, taper rate could be based on Radius, parent-daughter ratio on Path Distance, and bifurcation probability, branch path length, and daughter ratio all on Branch Order. With five basic parameters and three fundamental determinants, this creates an additional 3^{5} (minus the three “pure” cases) variants of this model (hence the name “243 Mix”). When comparing the individual and % Mix results to the more numerous 243 mix results, both the individual models and the % Mix models were run a total of 243 times with different random seeds.

Any morphometrics not directly used in the algorithm are “emergent” to the model. We chose four emergent morphometrics to compare virtual and real cells, selected for their biological and electrophysiological significance (_{1}−n_{2})/(n_{1}+n_{2}−2), where n_{1} and n_{2} are the number of terminal tips of the larger and smaller daughter subtrees, respectively. The total surface area is a size metric, while surface area asymmetry is defined by the same expression as above, but with n_{1} and n_{2} representing the surface areas of the daughter subtrees. Mean emergent morphometric values for each group of real trees are reported in the last four columns of

A custom java program (LNded2.0), running on a Pentium M under Windows XP, extracts the basic parameters from the real cells, fits them according to the appropriate fundamental determinants, and samples the resulting statistical distributions to create virtual dendrograms. The program then outputs the emergent morphometrics from real and virtual trees to Microsoft Excel for comparison and analysis. The code and necessary documentation for all model variants is available for public download under the ModelDB section

Error bars in all figures represent standard error unless otherwise noted. An asterisk directly above a column signifies a significant difference (P<.05) from the other two columns while an asterisk between two columns signifies a significant difference only between those two columns as determined by the Mann-Whitney U non-parametric comparison using