Wind loads and competition for light sculpt trees into self-similar structures

Abstract

Trees are self-similar structures: their branch lengths and diameters vary allometrically within the tree architecture, with longer and thicker branches near the ground. These tree allometries are often attributed to optimisation of hydraulic sap transport and safety against elastic buckling. Here, we show that these allometries also emerge from a model that includes competition for light, wind biomechanics and no hydraulics. We have developed MECHATREE, a numerical model of trees growing and evolving on a virtual island. With this model, we identify the fittest growth strategy when trees compete for light and allocate their photosynthates to grow seeds, create new branches or reinforce existing ones in response to wind-induced loads. Strikingly, we find that selected trees species are self-similar and follow allometric scalings similar to those observed on dicots and conifers. This result suggests that resistance to wind and competition for light play an essential role in determining tree allometries.

Full text

ARTICLE
Wind loads and competition for light sculpt trees
into self-similar structures
Christophe Eloy 1, Meriem Fournier2, André Lacointe3& Bruno Moulia3
Trees are self-similar structures: their branch lengths and diameters vary allometrically within
the tree architecture, with longer and thicker branches near the ground. These tree allo-
metries are often attributed to optimisation of hydraulic sap transport and safety against
elastic buckling. Here, we show that these allometries also emerge from a model that
includes competition for light, wind biomechanics and no hydraulics. We have developed
MECHATREE, a numerical model of trees growing and evolving on a virtual island. With this
model, we identify the fittest growth strategy when trees compete for light and allocate their
photosynthates to grow seeds, create new branches or reinforce existing ones in response to
wind-induced loads. Strikingly, we find that selected trees species are self-similar and follow
allometric scalings similar to those observed on dicots and conifers. This result suggests that
resistance to wind and competition for light play an essential role in determining tree
allometries.
DOI: 10.1038/s41467-017-00995-6 OPEN
1Aix Marseille Univ, CNRS, Centrale Marseille, F-13013 IRPHE Marseille, France. 2LERFoB, INRA, AgroParisTech, F-54000 Nancy, France. 3UCA, INRA, UMR
PIAF, F-63000 Clermont-Ferrand, France. Correspondence and requests for materials should be addressed to C.E. (email: eloy@irphe.univ-mrs.fr)
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Tree branching networks are generally self-similar. As a
result, the diameters and lengths of branches decrease with
the distance to the ground1. However, newly grown branch
segments always have approximately the same length2. The
observed hierarchy of lengths is in fact due to complex reconfi-
gurations through the continuous growth of new branches and
the pruning of old ones. Self-similarity is thus intimately linked to
the growth history of trees.
Self-similar properties of individual trees can be quantified by
measuring the radii, lengths and masses of each branch. Such
measurements have indicated that these quantities vary allome-
trically, as expected for self-similar structures3,4. Interestingly,
allometric scalings are also observed when comparing inter or
intraspecifically different populations of trees1,5,6. These scalings,
which are usually what is meant by ‘tree allometry’, relate tree
height, stem biomass, trunk diameter, etc.
Two classes of theoretical explanations have been given to
account for these allometric laws: mechanical4,7–9and hydrau-
lic10–12. Mechanical explanations date back to the work of
Metzger13, who proposed that wind-induced stresses should be
constant along trunks. This argument is related to the ‘axiom of
uniform stress’14,15, a necessary condition to minimise the
amount of material needed to support a load. Optimal mechan-
ical design is associated to the now well-established process of
thigmomorphogenesis, which is the plant response to mechanical
signals16,17. The ecological significance of thigmomorphogenetic
acclimation has long been recognised18, even if it has yet to be
implemented in ecological forest models.
Among the mechanical arguments, the concept of ‘elastic
similarity’has often been used4. Elastic similarity is an allometric
law that relates branch radii and lengths, such that the deflection
of the branch tip under self-weight is proportional to its length.
The same allometric law can be recovered for upright axes when a
constant safety factor against elastic buckling is enforced4.
Although it is generally admitted that wind loads offer a bigger
challenge to trees than buckling19,20, elastic similarity is the main
mechanical component of many allometric models5,21,22.
Within the hydraulic models, the pipe model10 or the initial
version of the West, Brown and Enquist (WBE) model23 have
been highly influential. In these models, a tree is modelled as a
fractal assembly of sap-conducting pipes. In its current version
however, the WBE model for plants21, as well as related
models5,22, also include a mechanical principle: trees are mod-
elled as volume-filling networks following the principle of elastic
similarity. With this approach, several allometric laws can be
deduced, relating trunk radius, tree height, stem biomass and leaf
biomass.
Mechanical and hydraulic models have often been compared,
opposed or combined9,11,24, but both rely on simplifications to
be questioned here. First, most models do not consider explicitly
the evolutionary mechanisms25. Second, tree architectures are
generally prescribed, without addressing growth. Therefore, they
cannot reflect the specific reconfiguration mechanisms found in
trees. Considering tree growth is the viewpoint of functional-
structural models26, which consider plants as an assembly of
individual organs, explicitly describing development and carbon
allocation27. These models, such as LIGNUM28, GREENLAB29,
AMAP30 or L-PEACH31 are usually based on a large number of
empirical parameters with the aim of modelling particular spe-
cies. An alternative approach is to exploit the recursive char-
acteristics of tree architectures by using the formal grammar of L-
systems32. With this approach, the self-organising processes
associated with growth, competition for light and interactions
with the environment can be addressed33. However, in these
models, evolutionary processes and wind biomechanics are
usually neglected.
To address the limitations of past approaches, we propose
MECHATREE, a new functional–structural model of tree growth.
This model integrates important biological processes related to
growth, architecture reconfiguration and evolution: competition
for light, carbon allocation, thigmomorphogenesis, wind-induced
pruning and genetic evolution. The main novelty of
MECHATREE lies in its ability to compute the growth and
evolution of entire ecosystems. We will use this feature to address
two important questions: What is the carbon allocation strategy
best adapted to competition for light, resistance to wind and
reproduction needs? Do the selected growth strategies yield tree
architecture and allometry consistent with empirical laws?
In MECHATREE, the modelling units are branch segments. In
contrast, models such as SORTIE34, ITD35 or SERA36 are indi-
vidual-based: the modelling units are individual trees with
species-specific allometric parameters. These models can address
the dynamics of a forest ecosystem, but are not suited to study the
origin of allometric scalings in individual tree architectures.
In the present study, we simulate a simplified version of evo-
lution on uniform virtual islands, with no gene influx at the
boundary. This island ecosystem is known to rapidly lead to a
single or very few dominant adapted species37. To speed up
genetic divergence between lines, we further assume autogamy
(child and parent share the same genome, except for slight
mutations). This island environment is submitted to realistic
recurrent wind gales38, and resource competition is limited to
light accessibility. In this simplified environment, in silico evo-
lution allow for a direct falsification of the selective force behind
allometric scalings. With this model, we find that self-similarity of
branch lengths and fractal dimension emerge through competi-
tion for light, while branch diameters are set through the response
to wind-induced stresses and eventually yield Leonardo’s rule of
area conservation.
Results
Structural units. Trees are modelled as modular structures of
different units: segments, foliages and seeds, together with a
carbon reserve. New units can be added, provided sufficient
photosynthates have been produced, and units can be pruned by
wind. Our aim is to mimic the main characteristics of an
angiosperm-like phenotypic set39. In building MECHATREE,
several simplifications have been made, with the goal of keeping
the model parsimonious and manageable. In particular, we have
neglected the selective pressure exerted by hydraulics through the
cost of transport and embolism. This is by no means because
hydraulics is not important, but we wanted to assess whether a
model based on light competition and wind-induced alone could
predict realistic allometries.
Tree branches are assemblies of segments, which are cylinders
of varying diameter d, but with always the same length L
Table 1 Parameters of MECHATREE
Parameter Symbol Value
Segment length LArbitrary
Twig diameter d
0
0.1L
Twig volume V
0
pLd2
0=4
Foliage diameter –L
Foliage transparency α
fol.
0.5
Cauchy number C
Y
2×10
−5
Volume produced by foliages V
prod.
4V
0
l
Maintenance thickness e0.02L
Forest radius R200L
Mutation probability p
mut.
0.05
Mutation amplitude δg0.005
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(Table 1). These segments are connected at their extremities, such
that each segment has a parent segment (except for the trunk),
and 0, 1 or 2 child segments. Segments with no child, called twigs,
are terminated by an assembly of leaves, called a foliage, modelled
as a sphere of diameter Lcentred on the segment distal end
(Fig. 1a). Seeds can be produced at the twig ends. The reserve,
whose exact location is not specified, stores assimilates from the
current year that will be mobilised the following year to support
primary and reproductive growth. Reserves have been included
because of their significant impact on tree capacity to recover
from major disturbances like strong wind damages.
Growth. The molecular regulations controlling growth strategies
are implemented using formal neural networks. Here, neural
networks are used as a tool that allows for an agnostic and flexible
modelling of complex physiological regulations that do not
involve actual neurons. Here, we make use of an important result
known as the ‘universal approximation theorem’, which states
that any continuous function can be approached with any pre-
scribed accuracy provided the number of hidden neurons is large
enough40. As illustrated in Fig. 1c, the artificial neural networks of
MECHATREE consist of three layers: an input layer, a hidden
layer and an output layer (for details, Methods section). Through
the neuronal coefficients, these neural networks relate function-
ally the inputs to the outputs.
Growth processes are divided into ‘primary growth’(the onset
of new segments and seeds), and ‘secondary growth’(the growth
in diameter of existing segments, Fig. 1b). The strategy of primary
growth is implemented with a 2–input, 3–hidden-neuron and
3–output neural network. The inputs are the volume of carbon
contained in the reserve and the number of foliages in the tree.
The outputs are a photosensitivity parameter and the proportions
of carbon allocated to new segments and seeds. When new
segments (children) of diameter d
0
=0.1Lare added at the distal
end of an existing segment (parent), geometrical rules inspired
from the seminal models of Honda41, and Niklas and Kerchner42
are used (Fig. 1d, Methods section).
Secondary growth is implemented with a 2–input, 3–hidden-
neuron and 1–output neural network. The inputs are the relative
wind-induced stress felt by the segment, σ
max
/σ
0
(Methods
section), and the number of foliages irrigating the segment. The
output is a safety factor Saccounting for the thigmomorphoge-
netic response (Methods section). The sink strength of a segment
is the sum of the volume needed to achieve a certain safety against
wind loads and a maintenance volume calculated as V
maint.
=
πLde, with e=0.02Lthe thickness of the outer layer to be renewed
every year. For every segment, this sink strength is equally
partitioned among the foliages situated above in the hierarchy:
each segment will ‘request’a equal amount of photosynthates to
the foliages above.
For each foliage, the photosynthates produced have a volume
V
prod.
=4V
0
l, where V0¼πLd2
0=4 is the volume of newly grown
twigs, and 0 ≤l≤1 is the intercepted light calculated with a ray-
tracing method43 (Methods section). If the photosynthate volume
produced by a foliage exceeds the total sink strength of the
segments situated below, each segment receives its share and the
leftover is stored in the reserve. On the contrary, if the volume
produced is not sufficient, each segment receives photosynthates
in proportion to its sink strength (Fig. 1b).
Tree growth is affected by both exogenous factors (wind,
shade), and endogenous factors (branching angles, neural
network coefficients). These endogenous factors are the ‘genes’
of a tree species, and together constitute its ‘genome’. In the
present case, there are 31 genes: 3 genes for the branching angles,
and 18 and 10 genes for the coefficients of the primary-growth
and secondary-growth neural networks respectively. These 31
genes are complemented with 3 ‘neutral marker genes’used for
visualisation purposes.
Within this model, branch fall is provoked by mechanical
loading, and this can occur along two different scenarios: either
an extreme wind event occurs and branches can fracture with a
probability described by a Weibull distribution (Methods
section); or the foliage sources cannot provide enough photo-
synthates to ensure the maintenance costs of a given branch and
Foliage
Segment
Reserve
Seed
45°
d
L
ØL
Structural units
Physiological regulations Branching
Primary growth Secondary growth
Reserve
Reserve
Growth Foliage
New
segmentsSeed
Strahler rank
1
2
3
Children
Parent
t
b
b2
b1
t1
b
b
t2
2
1


4
5
6
Output
layer
OutputsInputs
Hidden
layer
Input
layer 1
j
k
iIji Okj
ab
cde
Fig. 1 Principles of MECHATREE. aEach virtual tree is an assembly of different units: segments, foliages, seeds and a reserve. bPrimary growth relies on
the reserve to grow new segments and seeds. Secondary growth costs include maintenance in addition to diameter growth of each branch segment. c
Formal neural networks are used to model the biochemical regulation of growth. dThe growth of new segments follows a rule based on three angles: θ
1
,θ
2
and γ.eIllustration of Strahler ordering of branches (Methods section)
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this branch will weaken over time to the point where it will fall
down whatever the level of mechanical load.
Competition. The algorithm used to simulate the growth and
evolution of a forest is divided into the following steps (steps 1–7
constitute a yearly cycle).
Step 0: Initialisation. In a circle of radius R=200L, either
20,000 individuals with random genomes or 4000 individuals
with selected genomes are sown at random locations. At this
seeding stage, they are all formed of a single vertical segment of
diameter d
0
=0.1Land a reserve of volume 2V
0
.
Step 1: Light interception. The sunlight intercepted by each
foliage is calculated (Methods section).
Step 2: Stress calculation. The maximum bending stress, σ
max
,
is calculated in each segment (Methods section).
Step 3: Secondary growth. The photosynthates produced by
foliages are allocated to maintenance and diameter growth in
each segment.
Step 4: Pruning. A wind velocity is picked with random
orientation and speed Ufollowing an exponential distribution,
such that the return period of wind speeds exceeding by 50% the
average yearly maximum U
0
is 100 yrs38. The probability of
pruning is then given by a Weibull distribution (Methods
section).
Step 5: Death. Trees die when one of these two conditions is
realised: (i) their age is larger than 1000 yrs; (ii) their age is larger
than 6 yrs and their number of segments is less than 10.
Step 6: Primary growth. The carbon stored in the reserve is
allocated to grow seeds and new segments.
Step 7: Reproduction. Seeds fall with a 45° angle with the
vertical and form single-segment trees with the same genome as
their parent except for slight mutations (Methods section).
With the goal of identifying the best-adapted growth strategies
in a competitive environment, a single-elimination tournament is
run. During the first round, the growth and evolution of 32
different forests is simulated. Each forest is initialised with 20,000
trees with random genomes. The natural selection of a tree
phenotype (and genotype) within a forest is a ‘game’that yields a
single or few ‘winners’, i.e. species that dominates all others
(Fig. 2). Although trees of the same species have a common
ancestor, their genomes differ slightly, because of mutations at
each generation. After these simulations have been run for 10,000
yrs, the genomes of the 2000 oldest trees, the ‘winners’, are
collected in each of the 32 forests.
In the 2nd round, the growth of 16 forests is simulated. These
forests are now initialised with 4000 trees, composed of the
winners of two first-round games. After 20,000 yrs, the 2000
winners are again collected in each game. This operation is
repeated at each round until the Final is reached (Supplementary
Fig. 1). During the Final, the overall winning species are
identified. Among the initial 0.64 million random genomes, these
overall winners are the ‘fittest’: not only have they survived more
than 200,000 yrs in a competing environment, but the slight
mutations at each generation have allowed their genomes to
adapt. This highly simplified evolutionary process makes it
possible to reach a meaningful growth strategy without a priori
knowledge.
In the Final, after a transient, two different species eventually
coexist. We have performed simulations over more than 1 million
year, and there is no sign of one of these species becoming extinct.
These two species are associated to two different ecological
niches: the periphery and the interior of the island (Supplemen-
tary Fig. 1). In the Supplementary Discussion, we analyse how the
allocation strategies of these two species differ (see also
Supplementary Figs. 2and 3). However, when simulations are
ran with smaller islands (R=40Lor 100Linstead of R=200L),
after a few thousand years, only the periphery species remains.
For convenience, in the following we will refer to this periphery
species as the ‘fittest species’.
We will now determine if the trees selected by present model
exhibits allometric scalings similar to those observed empirically.
First, competition for light can be assessed through self-thinning,
which is the relation between stand density and average tree
biomass. Second, interspecific tree allometry can be examined for
the species that have survived the first 3000 yrs of simulation.
Finally, for the fittest species, self-similarity is assessed by
examining different allometric scalings within an architecture:
self-similar ratios, tapering law and area conservation.
10 yrs
1000 yrs 3000 yrs 10,000 yrs
104
103
102
101
100
0 2000 4000 6000 8000 10,000
Time (yrs)
# species
# trees
30 yrs 100 yrs 300 yrs
a
b
Fig. 2 Evolution of a forest on a virtual island. aExample of the evolution of a virtual forest over 10,000 yrs (Supplementary Movie 1). Initially, 20,000 trees
with random genomes are sown. Each circle is a tree whose centre is at the centre of gravity of foliages, and whose radius is the standard deviation of the
foliage distribution projected onto the ground. The outer black circle of radius R=200Lis the habitat limit. Colours are determined using three neutral
marker genes as RGB values. Because trees growing at the periphery are on average larger than the others, allometric statistics (see below) are performed
on trees with their trunk in a central zone of radius 0.9R.bNumber of trees and number of species as a function of time, for the same forest as in a
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Self-thinning. The empirical −3/2 self-thinning law for plant
populations states that the average biomass of plants decreases as
n−3/2, with nthe plant density44. This relation can be assessed by
computing how the ‘effective number’Nand ‘effective biomass’
Mvary with time45 (Methods section). In Fig. 3a, these quantities
are plotted for the initial 500 yrs of the first-round forests. Initi-
ally N¼2´104and M¼V00:008 L3, then, as trees grow, M
increases and Ndecreases following an allometric relation:
M/NβST , with β
ST
≈−1.418 (95% confidence interval, CI:
−1.421–1.416). The present model is thus in agreement with the
empirical self-thinning law (i.e. β
ST
≈−3/2), showing that, in
MECHATREE, competition for light, growth and mortality are
consistent with empirical observations for young forests.
Tree allometry. In Fig. 3b, trees found in first-round forests are
compared. Height H, crown radius C(measured as the standard
deviation of the foliage distribution projected onto the ground, as
in Fig. 2a), number of foliages Nand stem biomass Bare plotted
as a function of the trunk diameter d
trunk
for each tree. It shows
that the ~1000 different species that have survived the initial
3000 yrs exhibit allometric relations: H/dβH
trunk with β
H
≈0.87
(95% CI: 0.871–0.876), C/dβC
trunk with β
C
≈0.80 (95% CI:
0.802–0.805), N/dβN
trunk with β
N
≈1.97 (95% CI: 1.966–1.971),
B/dβB
trunk with β
B
≈2.82 (95% CI: 2.822–2.825). Similar expo-
nents are found when the forests are composed of the two finalist
species (Supplementary Fig. 4) and a sensitivity analysis shows
that these exponents depend only weakly on the model para-
meters (see below).
From these exponents, different experimental allometric
relations can be recovered. First, it can be seen that the biomass
roughly follows the classical scaling B/Hd2
trunk. Then, assuming
that the total mass of leaves, M
L
, scales as the number of foliages
N, ones finds that ML/BβML and ML/dβN
trunk, with β
ML
=0.68
and β
N
=1.97. These values are similar to the exponents
measured on angiosperms and gymnosperms: β
ML
=0.75 and
β
N
=2.175(they are about 9% smaller). In addition, the
distribution of trunk diameters in the forests composed of the
finalist species scales as d2
trunk (Supplementary Fig. 5). This scaling
is the same as the one generally observed in forest communities,
which has also been predicted by the theory of metabolic
ecology46,47.
These allometric scalings emerge in MECHATREE because the
trees selected through the single-elimination tournament share
common characteristics. All have a similar safety factor against
wind loads S≈3 (Supplementary Fig. 6c), and their architecture
is self-similar with a fractal dimension around D≈2.5 (Supple-
mentary Fig. 6e). Finally, in the literature, different values of the
allometric exponent have been reported in the interval 0.54 ≤
β
H
≤0.8948–50. The exponent β
H
=2/3 has also been predicted
based on arguments of elastic similarity4. However, some authors
argued that the relation between the logarithms of tree height and
trunk diameter is not linear but curvilinear because of finite-size
effects11,51,52. The same trend is observed in our data: β
H
tends
to decrease with d
trunk
(Fig. 3b). Another reason for curvilinearity
is that young trees are not self-similar (see below).
103
102
101
100
10–1
103
102
101
100
10–1
101102103
N
104
dtrunk
10–1 100101
Height
Size allometries vs. trunk diameterStand self-thinning
Slope: –1.42
Crown radius
# foliages
Biomass
2.82
0.87
0.80
Slope: 1.97
ba
M
Fig. 3 Allometric scalings. aFor the 32 forests of the first round, the effective biomass M(unit L3) is plotted as a function of the effective number of trees N,
for the first 500 yrs (the light blue line shows the history of a particular forest). Only 2% of the dataset is shown, but the red line shows a linear regression
fit on the entire dataset with Nas weight. Large excursions to the left of the regression line correspond to strong wind events during which a large number
of trees can die (Fig. 2b and Supplementary Movie 1). bEach tree in the 32 forests of the first round is extracted after 3000 yrs. Their height H(unit L),
crown radius C(unit L), number of foliages Nand stem biomass B(unit L3) are plotted as a function of their trunk diameter d
trunk
. For clarity, only 5% of the
trees are shown, but the solid lines show reduced major axis regression (RMA) on the whole dataset, with Nas weight. The dashed lines show the results
of the AMT model (see below)
Fig. 4 Partial view of the forest in the Final round on a small island
(R=40L), when only the fittest species remains. Only the largest tree has
been coloured for clarity. In this representation, the diameter of foliage
spheres is proportional to the light intercepted (Supplementary Movie 2)
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Self-similar ratios. In Fig. 4, a forest on a small island (R=40L)
is depicted when only the fittest species remains. Colouring the
branches of the tallest tree according to their Strahler ranks shows
qualitatively that the larger the rank, the thicker and the longer
the branches (Fig. 5a). A more quantitative analysis is performed
by examining the number of branches n
k
, their mean length l
k
,
diameter d
k
and cross-sectional area a
k
, as a function of their rank
k(Fig. 5b). These quantities follow a geometric progression such
that four self-similar ratios can be defined: the branching ratio R
n
=n
k
/n
k+1
, the length ratio R
l
=l
k+1
/l
k
, the diameter ratio
R
d
=d
k+1
/d
k
and the area ratio R
a
=a
k+1
/a
k
. A linear regression
shows that R
n
=3.51 (95% CI: 3.42–3.60), R
l
=1.60 (95% CI:
1.52–1.67), R
d
=1.85 (95% CI: 1.81–1.90) and R
a
=3.41 (95% CI:
3.23–3.61).
The existence of branching and length ratios, R
n
and R
l
, both
independent of k, means that the tree skeleton is self-similar and
that a fractal dimension can be defined: D=lnR
n
/lnR
l
=2.68 (95%
CI: 2.39–3.04). This fractal dimension is a measure of how the
tree structure fills space. As can be seen in Fig. 5c, fractal
dimension varies with time. When the tree is very young, Dis not
well defined, mainly because branch lengths do not progress
geometrically with Strahler order (see open symbols in Fig. 5b).
Then, as the tree grows and ages, Dfluctuates in the interval
2<D<3. Note that the amplitude of these fluctuations is much
lower for a tree growing without competitors (Supplementary
Figs. 7–10). It can be noted that R
l
≈R
d
, which means that branch
aspect ratio does not vary to a great extent (9 <l
k
/d
k
<25).
Another property is that R
n
≈R
a
, meaning that the total cross-
section of branches is independent of the rank. This rule of area
conservation can also be established with a different method (see
below).
Self-similar ratios and fractal dimensions have been rarely
measured in individual trees (Table 2). So far, the results of the
present model (3 <R
n
<4, 1.5 <R
l
<2 and 2 <D<3, Fig. 5c)
seem consistent with the available data. Not only the growth
strategy developed by the fittest virtual species yields a self-similar
architecture, but its quantitative characteristics resemble empiri-
cal observations.
6
1
2
3
4
5
7
8
Area ratio
〈 〉
0
1
2
3
4
5
Rn
RI
D
1
104
102
102
101
100
10–2 10–1 10010110–1
100
101
101102
100
# branches
Mean length
Mean area
Mean diameter
d
Branch tapering
Slope: 2/3 Slope: 3/2
Slope: 1
Strahler order Architectural self-similarity
Strahler rank
Mean: 0.94
Area conservation
Evolution of fractal dimension
Time (yrs)
100
100
10–2
2345678 0 200 400 600 800 1000
ab c
de
〈 〉
Fig. 5 Self-similarity. aRepresentation of the Strahler order of each branch for the tree illustrated in Fig. 4.bThe number of branches, their mean length
(unit L), area (unit L2) and diameter (unit L) are plotted as a function of their Strahler order for the 999-year-old tree illustrated in Fig. 4and in a. Error bars
show standard deviations around these means. Solid lines are regression fits on the first 7 ranks. Open symbols connected by dotted lines represent the
same quantities when the tree is only 25 years old. cEvolution of the branching ratio R
n
, length ratio R
l
and fractal dimension D=lnR
n
/lnR
l
during the
lifetime of the same tree. Grey bars show the 80% confidence interval for D.dBranch tapering is illustrated by plotting, for each segment, the average
distance from the foliages, ‘hi, as a function of the diameter d. The Strahler order of each segment is represented with the same colour code as in a.e
Assessment of Leonardo’s rule of area conservation across branching nodes. For each node, the area ratio (i.e. the ratio between the total cross-sectional
area of children segments and the parent area) is plotted as a function of the average distance ‘hiof the parent segment from the foliages. The average
area ratio measured for ‘
hi
>1:5 is 0.94. When restricted to ‘
hi
>10, the average is 0.985
Table 2 Self-similar ratios
R
n
R
l
R
d
D
Virtual tree in forest (Fig. 4) 3.51 1.60 1.85 2.68
Virtual tree alone
(Supplementary Fig. 7)
3.50 1.74 1.88 2.26
Red Oak43.83 –1.56 –
Poplar44.22 –1.86 –
Fir34.8 2.7 –1.6
Apple Tree68 4.35 –1.90 –
Birch tree68 4.00 –1.94 –
Pinyon pine69 3.63 1.71 1.81 2.40
The ratios R
n
,R
l
,R
d
and fractal dimension Dfor the virtual trees of Fig. 4and Supplementary
Fig. 7are compared with empirical data of the literature
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Tapering. Tapering describes, through an allometric relation,
how the diameter dof a segment is related to its distance ‘from
the branch apex: d/‘β4. In a ramified structure though, ‘is not
unique and is replaced by ‘
hi
, the average distance of all paths
connecting a segment to its descendant foliages. In Fig. 5d, ‘hiis
plotted as a function of d, for the tree of Fig. 4. This plot is very
similar to what has been measured on trees (e.g. Fig. 6 in ref. 4.).
It exhibits different regions: for small d, the scatter is important
and data do not follow a simple allometric relation; for inter-
mediate d, the allometric relation d/‘
hi
βwith β≈3/2 seems to
be reached; for large values of dcorresponding to the trunk, β≈
2/3, which is the exponent expected for uniformly distributed
wind-induced loads along the trunk height14. This is likely due to
the fact that the ‘sail area’of the trunk is of the same order as the
sail area of all foliages.
Area conservation. In his notebooks, Leonardo da Vinci
observed that ‘all the branches of a tree at every stage of its height
when put together are equal in thickness to the trunk’9. In other
words, according to Leonardo’s rule, the cross-sectional area
should be conserved across branching nodes on average. One way
to assess the validity of this rule is to plot for every branching
node, the area ratio (a
1
+a
2
)/a
0
, where a
1
and a
2
are the cross-
sectional areas of children branches and a
0
is the area of the
parent branch. A recent study has shown that the average value of
this ratio is between 0.90 and 1.05 for five species: Balsa, Maple,
Oak, Pinyon and Ponderosa pine53. In MECHATREE, the same
assessment of Leonardo’s rule has been performed and is shown
in Fig. 5e. This plot is very similar to the measurements made on
real trees (e.g. Fig. 3a in ref. 53.). In both cases, the average value
of the area ratio is close to 1, which is consistent with Leonardo’s
rule of area conservation.
Sensitivity analysis. By design, MECHATREE involves as few
parameters as possible for a process-based model (Table 1). Some
of these parameters, related to wind-induced loads, maintenance
costs and photosynthesis, have been chosen to match empirical
observations. In particular, the Cauchy number has been fixed to
CY¼ρU2
0=σ0¼2´105, with an air density ρ=1.2 kg m−3,an
average yearly maximal wind U
0
=40 m s−1and wood strength
σ
0
=100 MPa54. The maintenance thickness is taken equal to
e=0.02 L, or, by taking L=10 cm, is equal to e=2 mm, the
typical thickness of the inner bark. Taking a foliage with a leaf
surface of 150 cm255, which is about half the area of a sphere of
diameter L=10 cm, the optical transparency is set to α
fol.
≈0.5.
With a dry wood density of ρ
wood
=613 kg m−356,57,V
0
corre-
sponds to a dry mass of 4.8 g. Taking a typical net assimilation
rate of leaves of 10 g m−2per day58,59, each foliage produces
during 6 months about 27.3 g of photosynthates. This corre-
sponds to V
prod.
≈4V
0
l(assuming 70% of full light on average, i.e.
l=0.7).
To test the robustness of the allometric laws, we have
conducted a sensitivity analysis on four model parameters: C
Y
,
e,α
fol.
and V
prod.
. Model parameters have been varied, one at a
time, in a ±30% interval around their reference values given in
Table 1. Each time, 16 simulations are performed in forests of
radius R=200L. After 3000 yrs, the data of all trees are collected
and RMA regressions are performed to identify the allometric
exponents and intercepts (Fig. 3b), together with the typical safety
factor S, and an average fractal dimension D(see section
‘Sensitivity analysis on the allometric laws’in Methods section).
The main result is that, although trees grow to a larger size
when C
Y
or eare decreased, or when V
prod.
or α
fol.
are increased,
the allometric exponents β
N
,β
B
and β
ML
do not vary significantly.
The only exception is the dependence of β
H
, which is likely due to
a curvilinear relation between the logarithms of Hand d
trunk
(Fig. 3b). The allometric laws emerging from MECHATREE are
therefore robust to variations of the model parameters.
An interesting result of the sensitivity analysis is the
dependence of the fractal dimension on the foliage transparency.
This result is also confirmed by a parametric analysis that consists
in comparing how an isolated tree of the fittest species grow
when the model parameters are varied (Supplementary Methods).
Both analyses show that fractal dimension increases with
foliage transparency and this can be interpreted as follows. The
outer surface covered by foliage clusters generally has a
dimension around 2 (it can be slightly larger if it has some
fractal roughness). When foliage clusters are opaque, foliage
inside this surface does not intercept any light and will eventually
be shed because the branches supporting them do not have the
resources to ensure maintenance costs. Since foliages and the
structure supporting them have generally the same dimension, we
expect D≈2 for opaque foliages. This contrasts with the case of
fully transparent foliages, where the structure is expected to be
volume-filling (i.e. D=3). Owing to the central role of
chlorophyll in both photosynthesis and leaf transmittance
properties, V
prod.
and α
fol.
are likely to be negatively correlated.
Whether there is an optimal transparency remains, however, an
open question.
The AMT model. Genotypes that survive the initial 3000 yrs of
simulation exhibit allometric relations close to the ones observed
in nature (Table 3). Interestingly, these relations can be recovered
with a simple analytical model that we shall call the AMT
(Analytical MechaTree) model. The AMT model is thus a way to
capture the minimal set of factors explaining the emergence of
allometric scalings in MECHATREE. It is also useful to compare
our results with other models based on an analytical approach,
such as the WBE model21.
To build the AMT model, we use the three emergent results of
MECHATREE: (i) tree architectures (skeletons) are self-similar;
(ii) their fractal dimension is D≈2.5; (iii) their safety against
wind loads is constant. If the tree skeleton is self-similar,
branching and length ratios are independent of the rank k, and a
fractal dimension Dcan be defined such that Rn¼RD
l. Consider
now a branch at rank k. It is fed by the phloem sap flows coming
from Nk¼Rk1
nfoliages, which are at an average distance l
hi
k¼
LR
k
l1

=Rl1ðÞfrom the branch base. As a result, the relative
wind-induced bending stress is σk=σ0¼16α
πCYSfol:Nkl
hi
k=d3
k,
where αis an order 1 geometrical parameter that accounts for
the angle between wind and branches. If the safety factor Sis
constant (Assumption iii), branch diameters will be such that σ
k
/
σ
0
=S−3/2. With these arguments, the diameter ratio is found to
be Rd¼RDþ1ðÞ=3D
n, and the allometric exponents are (see Methods
section, for details).
βH¼3
Dþ1;βN¼3D
Dþ1;ð1aÞ
βB¼2Dþ5
Dþ1;βML ¼3D
2Dþ5ð1bÞ
Remarkably, within this simple analytical model based on
geometrical and mechanical arguments, allometric exponents
only depend on D. Using the value D=2.5 in the previous
analytic formulae (Assumption ii) yields allometric exponents in
good agreement with the results of the numerical simulations of
virtual forests (Table 3). This is consistent with the fractal
dimension 2 <D<3 that has been computed independently.
Comparison with measurements on angiosperms and
gymnosperms5,6,50 shows also an excellent agreement (Table 3,
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Fig. 6). Besides, the value of the fractal dimension (D≈2.5) is
compatible with measurements on tree structures (Table 2) and
tree crowns (2.13 <D
fol.
<2.7660), which should be similar to the
dimension of the branch architecture. Note that the AMT model
correctly predicts not only the allometric exponents, but also the
allometric intercepts (Fig. 6and Supplementary Discussion). Note
also that agreement is excellent over a large range of trunk
diameters or stem dry mass (Fig. 6). This is rather surprising
because the model should only provide realistic predictions for
trees that are large enough to have a properly defined fractal
dimension and small enough such that gravity loads remains
negligible.
Discussion
In this paper, a generic functional–structural model of tree, called
MECHATREE, has been developed. This model has been used to
address the best growth strategy of trees when they compete for
light and are subject to wind-induced loads. Thanks to its relative
simplicity, MECHATREE allowed us to simulate entire forests
over long periods of time. This feature has been exploited by
running a single-elimination tournament to identify the fittest
virtual species in this simplified environment. The self-similar
and allometric properties of this species have then been quantified
and found to be similar to those of existing tree species.
The existence of branching, length and diameter ratios shows
that tree architectures are self-similar (more rigorously, they are
self-affine since the different lengths can vary differently with
scale). Yet, the whole structure is an assembly of segments of
exact same length. Self-similarity is thus an emergent property
resulting from the complex architecture reorganisations that
occur during the lifetime of a tree through wind-induced pruning
and light- and wind-dependent growth. Two important results
can be deduced from MECHATREE. First, self-similarity of the
tree skeleton and the existence of a fractal dimension Dmainly
emerge from competition for light. As a result, Dis strongly
linked to foliage transparency. Second, for realistic probabilities of
extreme wind events, the safety factor against wind loads is
approximately constant. This efficient strategy to resist winds
leads to self-similarity of the branch diameters and Leonardo’s
rule of area conservation.
Classical allometric laws relating a tree’s trunk diameter to its
height, stem biomass or leaf biomass have been recovered with
the present numerical model and with a simplified analytical
model we developed, the AMT model. Alternative models, such
as the SERA model36 or WBE-related models5,21,22 can also
predict allometric exponents close to the ones measured on trees
(Table 3). Comparison with the present models is interesting, but
meaningful only when the nature of the models is similar.
The SERA model is an individual-based model with species-
specific parameters describing how an individual tree grows when
it competes for space and light36. From this model, allometric
relations emerge at the population level (Table 3). In that sense, it
bears some similarity with the present study. There are two major
differences between MECHATREE and SERA however:
MECHATREE does not involve species-specific parameters other
than those selected by the simulated evolution; and the modelling
units are the branch segments, which allows us to address the
Table 3 Comparison of allometric exponents
β
H
β
N
β
B
β
ML
Virtual forests 0.87 1.97 2.82 0.68
AMT model 0.86 2.14 2.86 0.75
WBE model 0.67 2 3 0.75
SERA model 0.86 1.98 2.66 0.74
Dicots, conifers 0.73 2.17 2.89 0.74
(95% CI) (0.71–0.76) (2.01–2.32) (2.71–3.14) (0.738–0.742)
Tree height H, leaf mass M
L
, stem biomass Band trunk diameter d
trunk
are related through allometric relations: H/dbH
trunk,ML/dbN
trunk,B/dbB
trunk and ML/BbML . The allometric exponents found for the
virtual forests, the AMT model (D=2.5), WBE model (D=3)21, the SERA model for angiosperms36 are compared to empirical observations on dicots and conifers5,6,50, given with their 95% confidence
intervals
10–2 10–1 100101
100
101
102
103104105
100
10–5
10–10
10–8 10–6 10–4 10–2 100102104
102
100
10–2
10–4
Woody species
LS regression
RMA regression
WBE model
AMT model
10–3 10–2 10–1 100
Leaf mass (kg)
Angiosperms
Gymnosperms
RMA regression
WBE model
AMT model
Leaf mass (kg)
Angiosperms
Gymnosperms
RMA regression
WBE model
AMT model
Tree height vs. trunk diameter Leaf dry mass vs. stem dry mass
Stem mass (kg)
Height (m)
Trunk diameter (m) Trunk diameter (m)
Leaf dry mass vs. trunk diameter
ab c
Fig. 6 Comparison of the AMT model with empirical data. aScaling of plant height as a function of stem diameter. The allometric data of ref. 48 for woody
species are compared to least square (LS) regression H¼20:7d0:538
trunk
48, reduced major axis (RMA) regression H¼21:4d0:73
trunk
50, WBE model H¼25 d0:67
trunk
and present AMT model H¼26:7d0:857
trunk (Methods section and Supplementary Discussion). bAllometric data on leaf dry mass vs. trunk diameter5are
compared to RMA regression ML¼166d2:17
trunk
5, WBE model ML¼247d2
trunk and AMT model ML¼202d2:14
trunk.cAllometric relations between leaf dry mass
and stem dry mass from more than 11,000 records6are compared to RMA regression ML¼0:113 MS0:746, WBE model ML¼0:176 MS0:75 and AMT model
ML¼0:124MS0:75
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origin of self-similarity in individual trees, which is not possible
in SERA.
The WBE model is an analytical model based on geometrical,
mechanical and hydraulic arguments. In particular, it assumes
that trees are volume-filling fractal networks (i.e. D=3). It can be
used to derive allometric relations that are very similar to the
predictions of the AMT model (Fig. 6, Supplementary Table 1
and Supplementary Discussion). It is thus impossible to refute
either approach based on allometry only. One way to achieve
falsification would be to have more data on the fractal dimension,
which should be possible with the development of Lidar-based
technologies61, or to directly assess the mechanistic bases of the
processes involved, e.g. through wind mechanosensing62.
In the WBE model21, or related models22, mechanics is
implemented by enforcing an elastically similar allometric rela-
tion between axis lengths and radii4(Supplementary Discussion).
In MECHATREE, the mechanical stresses and the species-specific
thigmomorphogenetic response are calculated for each segment
every year. Self-similarity and allometry are not imposed, but
emerge from thigmomorphogenesis and competition for light,
two major processes in temperate deciduous forests62,63.
The importance of the various hydraulic traits (water use
efficiency, embolism) and mechanical traits (wind hazards, self-
weight) may vary between species habitats and plant stages. In
some cases, hydraulic performance may be a major selective
pressure, whereas in others wind mechanical safety will be
dominant. Yet these different situations may not be identified in
broad range allometric data, since both selective pressures yield
similar allometries. We may however speculate that the presence
of both selective pressures can increase the speed of natural
selection. We believe that further insights could be gathered by
combining the biomechanically-based AMT model proposed in
the present paper with the hydraulic hypotheses of the initial
WBE model23 to have a better understanding of both mechanics
and hydraulics.
Methods
Formal neural networks. The artificial neural networks used to model agnostically
the growth strategies of different species consist of an input layer of neurons where
the stimuli arrive with intensity x
i
(normalised to be of order 1). These stimuli are
linearly combined and sent to a hidden layer of neurons receiving a signal y
j
, with
y
j
=∑
i
I
ji
x
i
. The hidden-layer neurons then perform a nonlinear transformation of
the signal: y′
j
=tanh(5y
j
) (except for one neuron not linked to the input layer and
sending a unit signal). Finally these signals are linearly combined and transmitted
to the output layer with intensity z
k
, with zk¼PjOkjy′
j(Fig. 1c).
Primary growth. Carbon is converted into primary growth through constructions
costs: a new segments costs its volume V
0
(Table 1) and a seed costs 5V
0
, which
accounts for the volume of its initial reserve, 2V
0
, the volume of the first sprout, V
0
,
and a supplementary cost for dissemination and germination (2V
0
).
From a practical viewpoint, each year, the primary growth neural network
computes for each tree three outputs: the proportions of carbon allocated to new
segments (0 ≤P
seg.
≤1) and to seeds (0 ≤P
seed
≤1), with P
seg.
+P
seed
≤1, and a
photosensitivity parameter, 0 ≤p≤1. If V
res.
is the volume of the reserve, a portion
of it is allocated to the construction of n
seg.
=floor(P
seg.
V
res.
/V
0
) new segments and
another portion to the construction of n
seed
=floor(P
seed
V
res.
/5V
0
) seeds. Both new
segments and seeds are located according to the photosensitivity parameter p: for p
=0, locations are picked at random among twigs; for p=1 the most lit foliages are
selected.
Light interception. To calculate the light lintercepted by each foliage, a ray-tracing
numerical method is used43. It is assumed that shadow cast by segments can be
neglected, and that sunlight is uniformly distributed in the upper hemisphere,
which is divided into 32 equal solid angles. For each solid angle, the whole forest is
rotated, such that zis along the mean direction of the solid angle. Foliages are then
ordered by descending zand the (x,y) position of each foliage is discretised onto a
grid of L×Lsquares. When different foliages belong to the same grid square, the
highest one receives 1/32 of light, the second highest α
fol.
/32, the third α2
fol:=32, etc.,
where α
fol.
=0.5 is the optical transparency of foliages43,55. This procedure is
repeated for each of the 32 solid angles and gives a good approximation of the total
light interception with a numerical scheme that scales as Nlog(N), with Nthe
number of foliages. Note that the slowest part of this algorithm is to sort foliage
heights, for which a quicksort algorithm is used64.
Branching angles. New child segments are added by following geometrical rules
based on three angles: θ
1
,θ
2
and γ9,41,42 (Fig. 1d). For a given parent segment
oriented in the direction of the unit vector tand normal to the unit vector b(for
the trunk, bis a random unit vector in the horizontal plane), two child segments of
length Lare constructed in the plane normal to b. Their tangential unit vectors t
1
and t
2
are obtained by rotating twith the angles θ
1
+ϵδθ and θ
2
+ϵδθ around b.
For a given tree, angles θ
1
and θ
2
are two constants, δθ =10°, and ϵis a normal
random variable with standard deviation 1. The normal vectors b
1
and b
2
defining
the next-generation branching planes are obtained by rotating bwith an angle γ+
ϵδθ around t
1
and t
2
, respectively.
Genome and mutation. During the lifetime of a tree, its genome does not vary.
However, the seeds produced by a tree have a mutated genome. Mutation rules are
as follows: with a probability p
mut.
=0.05 each gene gis replaced by g+ϵδg, with δg
=0.005 the amplitude of the mutation and ϵa random normal variable of unit
standard deviation. Naturally, if ghappens to be below 0 or above 1 after the
mutation, it is set to 0 or 1, respectively. The values of p
mut.
and δghave been
chosen to ensure that moderate mutations can occur over 10,000 yrs, the typical
time scale of simulations.
Wind-induced stress. The wind-induced stresses in a tree are calculated assuming
a uniform wind velocity Uu, where uis a horizontal unit vector. The wind-induced
force on each foliage is then Ffol:¼1
2ρU2Sfol:u;where ρis the air density and S
fol.
=
0.25L2is the ‘sail area’of foliages in strong winds. Here a drag coefficient of 1 has
been assumed without loss of generality. In addition, the wind exerts also a force on
each segment. If nis the unit vector normal to both the wind and the segment (i.e.
n¼t´u=t´ukk) the force exerted on each segment is Fsegment ¼
1
2ρU2dL t´u
kk
2n´t;where the drag coefficient is also taken to be 1, dand Lare
the diameter and length of the segment. This force is applied on the segment centre
of mass and its moment at the segment base is simply Msegment ¼1
2Lt´Fsegment.
Now each segment transmits the forces and moments applied at its extremity to
its base. If F
top
and M
top
are the sum of forces and moments at a segment top, force
and moment at the base are
Fbase ¼Fsegment þFtop;ð2aÞ
Mbase ¼Msegment þMtop þLt´Ftop:ð2bÞ
The moment at the base M
base
has a bending component of intensity
M¼Mbase ´t
kk
. The corresponding maximal bending (tensile and compressive)
stress occurs at the surface and is σ¼32
πM=d3. To compute the maximal bending
stress σ
max
sensed by segments, wind speed is assumed to be constant, U=U
0
, but
different orientations separated by 45° angles are considered.
Fracture probability. The probability of fracture of a given segment can be
modelled by a Weibull distribution to take into account volume effects65. This
probability reads
Pσ;VðÞ¼1exp V
V0
σ
σ0

m

;ð3Þ
where Vis the volume of the segment, σis the bending stress in the segment, σ
0
is
the strength of the material, V
0
is a reference volume taken to be the volume of new
twigs, and m=10 is the typical Weibull’s modulus for wood54. Since the wind-
induced bending stress is proportional to ρU2
0, with U
0
the average yearly-maximal
wind, the probability of fracture is a function of a dimensionless Cauchy number
CY¼ρU2
0=σ0:ð4Þ
In particular, Pdoes not depend on the typical size L. Considering an average
yearly-maximal wind of U
0
≈40 ms−1and a wood strength σ
0
≈100 MPa54, the
typical value of the Cauchy number is C
Y
=2×10
−5.
The safety factor Sused to implement the thigmomorphogenesis response is
such that the segments aims to reach a volume V=SV
fract.
, with V
fract.
the volume
giving σ
max
=σ
0
.
Strahler order. A topological rank can be assigned to each segment, following a
method originally developed by Strahler for river networks66. Within this frame-
work, a ‘branch’is defined as an assembly of contiguous segments of same rank.
The principle of this ordering scheme is to assign the rank 1 to all terminal
branches (i.e. assemblies of segments starting at the twigs and ending at the first
branching node). Supposing that these rank 1 branches are then removed, the new
terminal branches are assigned the rank 2, and so on (Fig. 1e). Although there are
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alternative choices of ordering scheme, we chose Strahler ordering because it allows
for a better assessment of self-similarity in an asymmetric branching structure4,67.
Assessment of the self-thinning rule. Using the average biomass to assess the
self-thinning rule induces a bias towards small trees. This is because the size
distribution contains a large number of very small trees. To avoid this bias, we
follow a method proposed by Adler45 and we use the effective number and the
effective biomass. The effective number Nis the inverse of the probability that two
units of mass taken at random from all trees come from the same tree. The effective
biomass Mis the average biomass weighted by the biomass itself. These quantities
are given by N¼B2
tot:=B2and M¼B2=Btot:, with B
tot.
the total biomass and B
2
the
second moment of the biomass distribution45. When all trees have the same size, N
is the number of trees and Mtheir biomass.
Principle of the AMT model. Consider a regular fractal tree skeleton of fractal
dimension Dand branching ratio R
n
. Because of the definition of D, the length
ratio is simply Rl¼R1=D
n. In this regular architecture, a branch of Strahler rank kis
fed by Nk¼Rk1
nfoliages situated above in the hierarchy and its length is
lk¼LRk1
l. Besides, the path length that connects the branch base to any des-
cendant foliage is
‘k¼lkþlk1þþl1¼LRk
l1
Rl1LRk
l
Rl1:ð5Þ
The bending moment at the base of this branch due to wind-induced loads in
the foliages is
Mk¼α
2ρU2Sfol:Nk‘k;ð6Þ
with α=α
1
α
2
<1 a geometrical parameter due to the fact that the distance between
the branch base and the foliages is α
1
times smaller than the path length, and that a
branch is not necessarily orthogonal to the wind (the projection of the force is then
α
2
times smaller than in the orthogonal case).
It follows that the bending stress σ
k
at the base of the branch is (see section
‘Wind-induced stress’in Methods section)
σk
σ0
¼16α
πd3
k
CYSfol:Nk‘k:ð7Þ
Then, since the safety factor Sof the different branches and different species
does not vary substantially (generally 2.5 ≲S≲4, Supplementary Fig. 6c), the
branch diameter is given by
d3
k¼16αS3
2
πCYSfol:
RlL
Rl1R
Dþ1
Dk1ðÞ
n;ð8Þ
which means that the diameter ratio is Rd¼RDþ1ðÞ=3D
nand dk¼d1Rk1
d, with d
1
given by (8) for k=1. For the particular case D=2.5, one finds that Rd¼R7=15
n,
Rl¼R2=5
nand Rd¼R7=6
lsuch that the aspect ratio l
k
/d
k
is almost constant. In
addition, the area varies as the square of the diameter, such that Ra¼R14=15
n, which
means that the total cross-sectional area is almost constant across ranks.
Assuming now that the whole tree counts KStrahler orders, the diameter of the
axes is given by (8). The trunk is a special case because it is always vertical and
orthogonal to the wind, and thus α1=3
2times larger than axes with random
orientations
dtrunk ¼d0
1RK1
d;with d0
1¼α1
3
2d1ð9Þ
Then, the height of the tree is approximately
H‘K¼RlL
Rl1
dtrunk
d01

3
Dþ1
;ð10Þ
thus giving an allometric exponent for the tree height vs. trunk diameter, βH¼3
Dþ1.
The total number of foliages is
N¼RK1
n¼dtrunk
d01

3D
Dþ1
;ð11Þ
such that the allometric exponent for Nis βN¼3D
Dþ1. The branch biomass volume is
B¼π
4d2
KlKþRnd2
K1lK1þRK1
nd2
1l1

;ð12Þ
¼πd2
1LR2
dRl
4R2
dRlRn

dtrunk
d01

2Dþ5
Dþ1
;ð13Þ
which gives an allometric exponent for the biomass βB¼2Dþ5
Dþ1. Finally, using (11)
and (13), the exponent for the leaf biomass is found to be βML ¼3D
2Dþ5.
Taking α1¼1
2,α2¼1
2,R
n
=3.5, D=2.5, S=3, Sfol:¼L2
4,C
Y
=2×10
−5,onefinds
R
l
=1.65, R
d
=1.79, which is coherent with independent measurements (Table 2in
the main text). One also finds d′
1¼0:0552Land Eqs. (10), (11)and(13)yield
H37:1Ldtrunk
L

0:857
;ð14Þ
N814 dtrunk
L

2:14
;ð15Þ
B33:9L3dtrunk
L

2:857
:ð16Þ
These allometric laws derived from the AMT model can be compared to the
scaling relations found in the trees growing in the first-round virtual forests
simulated with MECHATREE (Fig. 3b). Except for the number of foliages, which is
somewhat over-predicted for large trees by this simple analytical model, there is a
remarkable agreement with the data extracted from the simulated trees, thus
proving that the arguments outlined above on mechanics and self-similarity
capture well most of the mechanisms affecting allometry in MECHATREE (same is
true for forests sown with only the two finalist species: Supplementary Fig. 4b).
To go further, one can try to compare the allometric scalings of the AMT model
to the data measured empirically. To do so, we assume a segment length L=10 cm,
a dry wood density ρ
wood
=613 kg m−3and the mass of one foliage m
fol.
=1.8 g.
Taking these values and using Eqs. (10), (11) and (13), the following relations are
obtained
H26:7d0:857
trunk;ð17Þ
ML202d2:14
trunk;ð18Þ
ML0:124M0:75
S;ð19Þ
where M
S
is the dry mass of stems and SI units have been used (metres for lengths
and kilograms for weights). These relations show excellent agreement with
allometric relations found in the literature (Fig. 6).
Sensitivity analysis on the allometric laws. We want to assess how the laws of
tree allometry depend on the model parameters. To perform this sensitivity ana-
lysis, we have modified the Cauchy number C
Y
, the maintenance thickness e, the
volume produced by foliages V
prod.
and the foliage transparency α
fol.
, one-factor-at-
a-time, around the reference values given in Table 1. Each allometric law can be
written as αXzβX, where α
X
is the intercept and β
X
is the exponent of the allometric
relation. Both of these constants a priori depend on the model parameters. To
characterise this dependency, we estimate the relative variation of each allometric
constant for relative variation of each parameter independently, in other words the
sensitivity
sβX;p¼p
βX

ref:
∂βX
∂p

ref:
;ð20Þ
where pis one of the model parameter (with this definition, the sensitivities s
a,b
are
always dimensionless).
To estimate the sensitivities, defined by Eq. (20), we have run 16 forest
simulations during 3000 yrs, for each values of the model parameters. The
macroscopic characteristics of each tree in these simulations (height, biomass,
trunk diameter, etc.) are then extracted and used to perform a standard major axis
regression similar to what has been done for the first-round forests (Fig. 3b). This is
repeated for 6 different values of each of the 4 model parameters, resulting in a total
of 384 simulations (taking ~1500 CPU-hours). The allometric constants calculated
from these simulations are plotted in the Supplementary Fig. 11. The sensitivities
are then estimated through the slope of the linear regression fit of these plots
(Supplementary Table 2). Examining the values of the sensitivities in
Supplementary Table 2, it appears that the allometric exponents β
X
do not vary
significantly with the model parameters. There is only one exception: when the
maintenance thickness is increased, or the volume produced by foliages decreased,
β
H
increases. This occurs because, in these cases, trees are much smaller on average
(see last column of Supplementary Fig. 11). As it has already been discussed in the
main text, the relation between the logarithms of the trunk diameter and the tree
height is not linear but curvilinear (probably because of finite-size effects). It results
that a fit made on small trees tend to overestimate β
H
. It thus explains the
sensitivities sβH;eand sβH;Vprod:observed.
For the sensitivities on the allometric intercepts, a similar conclusion can be
drawn: the dependencies on C
Y
and α
fol.
can be interpreted with the AMT model
(see below), and there is no significant dependency on eand V
prod.
, with the
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exception of α
N
, for which the sensitivities are about ±0.5. One explanation for this
specific behaviour may be that again when increasing eor decreasing V
prod.
, less
photosynthates are available for primary growth and thus trees are much smaller.
There is also a selection of a lower safety against wind loads and a higher fractal
dimension. All these effects concur to increase the number of leaves for a given
trunk diameter and, as a consequence, increase α
N
, but further research is clearly
needed to fully understand this effect.
From the analytic model for tree allometry (the AMT model), sensitivities can
also be calculated (Supplementary Table 3). To do so, we have assumed that fractal
dimension Ddepends linearly on the foliage transparency α
fol.
:D=2+α
fol.
,asitis
suggested from the parametric analyses (Supplementary Methods and
Supplementary Fig. 12). For allometric intercepts, agreement with the numerical
simulation is good: signs and orders of magnitude of the sensitivities are correctly
recovered both for C
Y
and for α
fol.
. For the allometric exponents, agreement is
excellent, showing that the simple analytic AMT model correctly incorporates the
principal mechanisms at play in setting the allometric laws in MECHATREE. It
also shows that the hypothesis of linear dependence of fractal dimension on foliage
transparency (D=2+α
fol.
) is coherent with the sensitivities observed.
Supplementary Table 4shows how fractal dimension, average tree size and
safety are sensitive to the model parameters. It should be noted that the tree species
used to compute these sensitivities have only evolved during 3000 yrs. On such a
short period, only rudimentary selection has had enough time to act. Yet, safety
against wind loads depend significantly on the model parameters. Our
interpretation of this dependence is the following. As expected, trees can grow
larger on average when more photosynthates can be allocated to primary growth
(either when C
Y
or edecreases, or when V
prod.
or α
fol.
increases). However, trees
that grow larger also take more time to grow. For them, a higher safety against
wind loads may be necessary to survive very rare wind events. For these large trees,
the fractal dimension also tends to be smaller because of finite-size effects. These
arguments explain the signs of all sensitivities in Supplementary Table 4, except
one: sD;αfol:. For larger transparencies of foliages, the fractal dimension tends to
increase. The explanation for this particular dependence is given above: for opaque
foliages, we expect a fractal dimension D=2 and for fully transparent foliages, we
expect volume-filling D=3.
Data availability. The data sets generated during and/or analysed during the
current study are available from the corresponding author on reasonable request.
Received: 8 April 2016 Accepted: 8 August 2017
References
1. Niklas, K. J. Plant Allometry: The Scaling of Form and Process (University of
Chicago Press, 1994).
2. Barthélémy, D. & Caraglio, Y. Plant architecture: a dynamic, multilevel and
comprehensive approach to plant form, structure and ontogeny. Ann. Bot. 99,
375–407 (2007).
3. Leopold, L. B. Trees and streams: the efficiency of branching patterns. J. Theor.
Biol. 31, 339–354 (1971).
4. McMahon, T. A. & Kronauer, R. E. Tree structures: deducing the principle of
mechanical design. J. Theor. Biol. 59, 443–466 (1976).
5. Enquist, B. J. & Niklas, K. J. Global allocation rules for patterns of biomass
partitioning in seed plants. Science 295, 1517–1520 (2002).
6. Poorter, H. et al. How does biomass distribution change with size and differ
among species? An analysis for 1200 plant species from five continents. New
Phytol. 208, 736–749 (2015).
7. Moulia, B. & Fournier-Djimbi, M. in Proc. of the 2nd Plant Biomechanics
Conference, Centre for Biomechanics, University of Reading, Reading, United
Kingdom (eds Jeronimidis, G. & Vincent, J. F. V.) 43–55 (Reading University,
Reading, 1997).
8. Bertram, J. E. A. Size-dependent differential scaling in branches: the mechanical
design of trees revisited. Trees 3, 241–253 (1989).
9. Eloy, C. Leonardo’s rule, self-similarity and wind-induced stresses in trees.
Phys. Rev. Lett. 107, 258101 (2011).
10. Shinozaki, K., Yoda, K., Hozumi, K. & Kira, T. A quantitative analysis of plant
form–the pipe model theory. I. basic analyses. Jpn. J. Ecol. 14,97–105 (1964).
11. Niklas, K. & Spatz, H. Growth and hydraulic (not mechanical) constraints
govern the scaling of tree height and mass. Proc. Natl Acad. Sci. USA 101,
15661–15663 (2004).
12. McCulloh, K. A. & Sperry, J. S. Murray’s Law and the Vascular Architecture of
Plants 85–100 (Taylor & Francis, Boca Raton, 2006).
13. Metzger, K. Der wind als massgeblicher faktor für das wachstum der bäume.
Münden. Forstl. Hefte 3,35–86 (1893).
14. Morgan, J. & Cannell, M. G. R. Shape of tree stems–a re-examination of the
uniform stress hypothesis. Tree. Physiol. 14,49–62 (1994).
15. Mattheck, C. Teacher tree: the evolution of notch shape optimization from
complex to simple. Eng. Frac. Mech. 73, 1732–1742 (2006).
16. Moulia, B., Coutand, C. & Lenne, C. Posture control and skeletal mechanical
acclimation in terrestrial plants: Implications for mechanical modeling of plant
architecture. Am. J. Bot. 93, 1477–1489 (2006).
17. Hamant, O. Widespread mechanosensing controls the structure behind the
architecture in plants. Curr. Opin. Plant. Biol. 16, 654–660 (2013).
18. Ennos, A. R. Wind as an ecological factor. Trends Ecol. Evol. 12, 108–111 (1997).
19. de Langre, E. Effects of wind on plants. Annu. Rev. Fluid Mech. 40, 141–168
(2008).
20. Albrecht, A. et al. Comment on “critical wind speed at which trees break”.Phys.
Rev. E 94, 067001 (2016).
21. West, G., Brown, J. & Enquist, B. A general model for the structure and
allometry of plant vascular systems. Nature 400, 664–667 (1999).
22. Savage, V. et al. Hydraulic trade-offs and space filling enable better predictions
of vascular structure and function in plants. Proc. Natl Acad. Sci USA 107,
22722–22727 (2010).
23. West, G. B., Brown, J. H. & Enquist, B. J. A general model for the origin of
allometric scaling laws in biology. Science 276, 122–126 (1997).
24. Niklas, K. J. & Spatz, H. C. Wind-induced stresses in cherry trees: evidence
against the hypothesis of constant stress levels. Trees 14, 230–237 (2000).
25. Bornhofen, S. & Lattaud, C. Competition and evolution in virtual plant
communities: a new modeling approach. Nat. Comput. 8, 349–385 (2009).
26. Guo, Y., Fourcaud, T., Jaeger, M., Zhang, X. & Li, B. Plant growth and
architectural modelling and its applications. Ann. Bot. 107, 723–727 (2011).
27. Lacointe, A. Carbon allocation among tree organs: a review of basic processes
and representation in functional-structural tree models. Ann. For. Sci. 57,
521–533 (2000).
28. Perttunen, J., Sievänen, R. & Nikinmaa, E. LIGNUM: a model combining the
structure and the functioning of trees. Ecol. Model. 108, 189–198 (1998).
29.Yan,H.P.,Kang,M.Z.,DeReffye,P.&Dingkuhn,M.Adynamic,architectural
plant model simulating resource-dependent growth. Ann. Bot. 93, 591–602 (2004).
30. de Reffye, P., Fourcaud, T., Blaise, F., Barthélémy, D. & Houllier, F. A
functional model of tree growth and tree architecture. Silva Fenn. 31, 297–311
(1997).
31. Allen, M. T., Prusinkiewicz, P. & DeJong, T. M. Using L-systems for modeling
source–sink interactions, architecture and physiology of growing trees: the
L-PEACH model. New. Phytol. 166, 869–880 (2005).
32. Prusinkiewicz, P. & Lindenmayer, A. The Algorithmic Beauty of Plants
(Springer-Verlag, 1990).
33. Palubicki, W. et al. Self-organizing tree models for image synthesis. ACM
Trans. Graph. 28, 58:1–58:10 (2009).
34. Pacala, S. et al. Forest models defined by field measurements: estimation, error
analysis and dynamics. Ecol. Monogr. 66,1–43 (1996).
35. Purves, D. W., Lichstein, J. W. & Pacala, S. W. Crown plasticity and
competition for canopy space: a new spatially implicit model parameterized for
250 north american tree species. PLoS ONE 2, e870 (2007).
36. Hammond, S. T. & Niklas, K. J. Emergent properties of plants competing in
silico for space and light: Seeing the tree from the forest. Am. J. Bot. 96,
1430–1444 (2009).
37. MacArthur, R. H. & Wilson, E. O. An equilibrium theory of insular
zoogeography. Evolution 17, 373–387 (1963).
38. Morton, I. D., Bowers, J. & Mould, G. Estimating return period wave heights
and wind speeds using a seasonal point process model. Coast. Eng. 31, 305–326
(1997).
39. Maynard Smith, J. Optimization theory in evolution. Annu. Rev. Ecol. Evol.
Syst. 9,31–56 (1978).
40. Cybenko, G. Approximation by superpositions of a sigmoidal function. Math.
Control Signals Syst. 2, 303–314 (1989).
41. Honda, H. Description of the form of trees by the parameters of the tree-like
body: effects of the branching angle and the branch length on the shape of the
tree-like body. J. Theor. Biol. 31, 331–338 (1971).
42. Niklas, K. J. & Kerchner, V. Mechanical and photosynthetic constraints on the
evolution of plant shape. Paleobiology 10,79–101 (1984).
43. Sinoquet, H., Thanisawanyangkura, S., Mabrouk, H. & Kasemsap, P.
Characterization of the light environment in canopies using 3D digitising and
image processing. Ann. Bot. 82, 203–212 (1998).
44. White, J. & Harper, J. L. Correlated changes in plant size and number in plant
populations. J. Ecol. 58, 467–485 (1970).
45. Adler, F. R. A model of self-thinning through local competition. Proc. Natl
Acad. Sci USA 93, 9980–9984 (1996).
46. Enquist, B. J. & Niklas, K. J. Invariant scaling relations across tree-dominated
communities. Nature 410, 655–660 (2001).
47. White, E. P., Ernest, S. K. M., Kerkhoff, A. J. & Enquist, B. J. Relationships
between body size and abundance in ecology. Trends Ecol. Evol. 22, 323–330
(2007).
48. Niklas, K. J. The scaling of plant height: a comparison among major plant
clades and anatomical grades. Ann. Bot. 72, 165–172 (1993).
49. Niklas, K. J. Reexamination of a canonical model for plant organ biomass
partitioning. Am. J. Bot. 90, 250–254 (2003).
NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00995-6 ARTICLE
NATURE COMMUNICATIONS |8: 1014 |DOI: 10.1038/s41467-017-00995-6 |www.nature.com/naturecommunications 11
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50. Niklas, K. J., Cobb, E. D. & Marler, T. A comparison between the record height-
to-stem diameter allometries of pachycaulis and leptocaulis species. Ann. Bot.
97,79–83 (2006).
51. Niklas, K. J. Size-dependent allometry of tree height, diameter and trunk-taper.
Ann. Bot. 75, 217–227 (1995).
52. Savage, V. M., Deeds, E. J. & Fontana, W. Sizing up allometric scaling theory.
PLoS Comput. Biol. 4, e1000171 (2008).
53. Bentley, L. P. et al. An empirical assessment of tree branching networks and
implications for plant allometric scaling models. Ecol. Lett. 16, 1069–1078
(2013).
54. Thelandersson, S. & Larsen, H. J. (eds.). Timber Engineering (Wiley, 2003).
55. Da Silva, D., Boudon, F., Godin, C. & Sinoquet, H. Multiscale framework for
modeling and analyzing light interception by trees. Multiscale Model. Simul. 7,
910–933 (2008).
56. Chave, J. et al. Towards a worldwide wood economics spectrum. Ecol. Lett. 12,
351–366 (2009).
57. Zanne, A. E. et al. Data from: towards a worldwide wood economics spectrum.
Dryad Digital Repository http://dx.doi.org/10.5061/dryad.234 (2009).
58. Poorter, H. & Remkes, C. Leaf area ratio and net assimilation rate of 24 wild
species differing in relative growth rate. Oecologia 83, 553–559 (1990).
59. McDonald, A. J. S., Lohammar, T. & Ingestad, T. Net assimilation rate and
shoot area development in birch (betula pendula roth.) at different steady-state
values of nutrition and photon flux density. Trees 6,1–6 (1992).
60. Zeide, B. & Pfeifer, P. A method for estimation of fractal dimension of tree
crowns. Forest Sci. 37, 1253–1265 (1991).
61. Raumonen, P. et al. Fast automatic precision tree models from terrestrial laser
scanner data. Remote Sens. 5, 491–520 (2013).
62. Bonnesoeur, V., Constant, T., Moulia, B. & Fournier, M. Forest trees filter chronic
wind-signals to acclimate to high winds. New. Phytol. 210,850–860 (2016).
63. Hautier, Y., Niklaus, P. A. & Hector, A. Competition for light causes plant
biodiversity loss after eutrophication. Science 324, 636–638 (2009).
64. Sedgewick, R. Implementing quicksort programs. Commun. ACM. 21, 847–857
(1978).
65. Bažant, Z. P. & Planas, J. Fracture and Size Effect in Concrete and Other
Quasibrittle Materials (CRC Press, 1998).
66. Strahler, A. N. Dynamic basis of geomorphology. Bull. Geol. Soc. Am. 63,
923–938 (1952).
67. Turcotte, D. L., Pelletier, J. D. & Newman, W. I. Networks with side branching
in biology. J. Theor. Biol. 193, 577–592 (1998).
68. Barker, S. B., Cumming, G. & Horsfield, K. Quantitative morphometry of the
branching structure of trees. J. Theor. Biol. 40,33–43 (1973).
69. Tausch, R. J. A structurally based analytic model for estimation of biomass and
fuel loads of woodland trees. Nat. Resour. Model 22, 463–488 (2009).
Acknowledgements
We kindly thanks Anne Atlan, Benjamin Audit, Eric Badel, Hugues Chaté, Yves Couder,
Catherine Coutand, Thierry Fourcaud, Xavier Leoncini and Agnès Schermann, for sti-
mulating and helpful discussions. This work was granted access to the HPC resources of
Aix-Marseille Université, funded by the project Equip@Meso (ANR-10-EQPX-29-01) of
the programme ‘Investissements d’Avenir’supervised by the ANR.
Author contributions
All authors conceived the study. C.E. wrote the code, carried out the simulations and
wrote the first draft. All authors contributed to the revisions of the manuscript.
Additional information
Supplementary Information accompanies this paper at 10.1038/s41467-017-00995-6.
Competing interests: The authors declare no competing financial interests.
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