Etude de la morphométrie des arbres par combinaison de la

géométrie fractale et de la physique statistique

J. Duchesne1, P. Raimbault2 and C. Fleurant1

1. UMR 105 Paysages et biodiversité

2. UMR SAGAH

Une loi universelle de la morphométrie des arbres ?

• Introduction

• Démonstration de la loi

• Résultats et discussion

• Conclusion

Une loi universelle de la morphométrie des arbres ?

• Introduction

• Démonstration of the law

• Résultats et discussion

• Conclusion

Réseaux hydrographiques

Arbres

Structures fractales ramifiées

Ont en commun l’invariance d’échelle

22

2

2

2

2

2

2

3

3

4

N1 : nombre de tronçons du 1er ordre

N2 : nombre de tronçons du 2ème ordre

………..

Rapports N1/ N2, N2/ N3, … Ni/ Ni+1 sont constants et notés

RC : rapport de bifurcation

Ni: nombre de tronçons d’ordre i

………..

Les rapports L2/ L1, L3/ L2, … Li+1/ Li sont constants et notés

RL : rapport de longueur

2L : longueur moyenne des tronçons d’ordre 2

1L : longueur moyenne des tronçons d’ordre 1

nL : longueur moyenne des tronçons d’ordre n

Ces deux résultats sont la marque d’une

structure fractale ramifiée

Une loi universelle de la morphométrie des arbres ?

• Introduction

• Démonstration of the law

• Résultats et discussion

• Conclusion

Nous proposons

d’utiliser un raisonnement

de physique statistique

Une loi universelle de la morphométrie des arbres ?

• 1. Introduction

• 2. Démonstration de la loi– Choix de l’espace symbolique

Symbolic space of Maxwell :the space of speeds

vx

vyvz

dvx

dvy

dvz

zyx

zyx dvdvdvN

NdvvvF

...,,

3

d3N is the number of molecules which the speed vector

ends to the elementary volume dvx dvy dvz , among a total number of molecules N

The two hypotheses of Maxwell

• the independence of the 3 speed components ;

• the isotropy of the speed directions

The independence of the 3 speed components involves :

)()()(,, 321 zyxzyx vfvfvfvvvF

So, the 3 variables are separated

The isotropy of the speed directionsis a natural hypothesis because

one can hardly imagine that some directions be privilegied

The distributions f1, f2, f3 have the same form :

)()()(,, zyxzyx vfvfvfvvvF

are sufficient conditions to determine the function F(v)

The two hypotheses of Maxwell

Analogy between thermodynamics and natural

networks• thermodynamics • natural networks

Maxwell approach Our approach

Analogy ...• thermodynamics • natural networks

Notion of speed vector module Notion of hydraulic length

v L

2222zyx vvvv nllllL 321

vx

vy

vz

dvx

dvy

dvz

2xv

2yv 2

zv

v

2vMaxwell approach

Analogy …thermodynamics natural networks

L = l1 + l2 + l3Our approach +… + ln

Analogy …thermodynamics natural networks

Analogy …thermodynamics natural networks

There are two differences

between the two approaches ...

Analogy …thermodynamics natural networks

• In thermodynamics • In natural networks

first difference :

there are 3 components there are n components

Analogy …thermodynamics natural networks

2v L

xv 1l

yv 2l

zv 3l

nl• • •

xvyv

zv

1l2l

3l

Analogy …thermodynamics natural networks

nl

Analogy …thermodynamics natural networks

• In thermodynamics • In natural networks

second difference :

the 3 componentshave the same mean

the n componentshave not the same mean

zyx vvv 2

123

12

ll

l

RlRll

Rll

Analogy …thermodynamics natural networks

2v L

xv1l

• • • • • •

Analogy …thermodynamics natural networks

xv 1l

yvlR

l2

zv 23

lR

l

1nl

n

R

l

• • •

xvyv

zv

1l lR

l2

23

lR

l

Analogy …thermodynamics natural networks

1nl

n

R

l

Maxwell’s two hypotheses

• the independence of the 3 speed components ;

• the isotropy of the speed directions

become

• the independence of the n length components ;• the isotropy of the directions of the symbolic space

The Maxwell function

)()...()(,,1

21321

nl

n

l R

lf

R

lflflllF

becomes

)()()(,, zyxzyx vfvfvfvvvF

The Maxwell results for f

)exp(2

...

11

2

1

321

il

i

il

ii R

l

R

lAlf

lflflf

become

)21

exp(

2

kT

mvkvf

x

x

And the same for vx, vy, vz

become

Une loi universelle de la morphométrie des arbres ?

• Introduction

• Démonstration of the law

• Résultats et discussion

• Conclusion

… it is necessary to respect two conditions

which are strongly related to the statistical physics :

1. the size of the system much be very large compared with

the elementary constituent which will be taken into account

2. the local properties of the system must be homogeneous

A large number of elementary constituents

Homogeneity of the population

Results with a population of trees

The population : 12 apple trees

4 years old

grown from the same parents

Order i RB RL

2 12.06 2.163 11.5 4.25

Theoretical 11.78 3.03

0200

400600

8001000

12001400

1600

1800

2000

0 20 40 60 80 100 120 140

li/rli-1

pdf(li)

Order 1

Order 2

Order 3

1. The number of hydraulic lengths,corresponding to apexes, cannot exceed a few thousand for a given class ;

2. Moreover, the distribution of hydraulic lengths, as well as the distribution of their n components can be more or less influenced by the environment constraints.

Results with a Cupressocyparis

In the same way,

RB and RL are calculated for all branches of the tree

Mean hydraulic length :

the average of all the hydraulic lengths of the tree

Order n :

the maximum order observed in the tree

0

200

400

600

800

1000

1200

1400

0 5 10 15 20 25

Ordre 1

Ordre 2

Ordre 3

Ordre 4(th.)

Série5

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

L

pd

f Experimental

Theoretical

Une loi universelle de la morphométrie des arbres ?

• Introduction

• Démonstration de la loi

• Résultats et discussion

• Conclusion

Courbes polygonum

0

50

100

150

200

250

300

350

400

0 10 20 30 40 50 60

l1

l2 réduit

Série3

Série4

Réseaux sur Titan

(source : ESA)

Biblio sommaire• Fleurant C., Duchesne, J., Raimbault, P., 2004. An allometric

model for trees. Journal of Theorical Biology, 227, 137-147.• Cudennec C., Fouad Y., Sumarjo Gatot I. & Duchesne J. 2004. A

geomorphological explanation of the unit hydrograph concept. Hydrological processes, 18, 603-621.

• Duchesne J., Raimbault P. & Fleurant C. 2002. Towards a universal law of tree morphometry by combining fractal geometry and statistical physics, in Proceeding "Emergent Nature", 7th multidisciplinary conference, M. M. Novak (ed.), World Scientific 2002, pp.93-102.

• Roland B. 2002. An attempt to characterise Hedgerow lattice by means of fractal geometry, , in Proceeding "Emergent Nature", 7th multidisciplinary conference, M. M. Novak (ed.), World Scientific 2002, pp. 103-112.

• Duchesne, J, Fleurant, C. & Capmarty-Tanguy, F., inventeurs, 2002, Procédé d’implantation de végétaux, plan d’implantation de végétaux obtenu et système informatique pour l’élaboration d’un tel plan, déposé à l’INPI le 25 juin 2002.

Merci de votre attention

Order ii

icN

NR 1

1

i

iLL

LR

1 2 12.06 2.16

2 3 11.5 4.25

Theoretical 11.78 3.03

zyx

zyx dvdvdv

NdvvvNF

..,,

3

This is the density of the points representing the speeds

zyxzyx dvdvdvvvvFN

Nd,,

3

As we have :

1,, zyxzyx dvdvdvvvvF

NNd 3