A stochastic description of branching structures of trees

J. theor. Biol. (1985) 112, 667-676

A Stochastic Description of Branching Structures of Trees MASAHIRO A G U t AND YOTA YOKOI~

t Department of Electronics, Faculty of Engineering, lbaraki University, Hitachi 316, lbaraki, Japan and ~t Department of Biology, Faculty of Science, lbaraki University, Mito 310, Japan (Received 21 February 1984) A systematic method of the description of the temporal growth of trees is presented by defining the age of each branch as the number of bifurcation times occurring in the development process from original ancestor branch to the progeny branch. It is shown that the Fibonacci branching, in which the population of terminal branches increases in the course of time following the Fibonacci number series, is a direct consequence of the apical dominance phenomenon in branch growth. Stochastic nature of branching is also introduced through taking the random bifurcation rate of branch into account. The computer simulations of the age construction rates of the stochastic description are found to be in good agreements with the observed data on the real trees such as Celtis sinensis Pers. var. Japonica, Zelkova serrata Makino or Cedrus deodara Loud.

1. Introduction

The description of static structures of trees has long been studied as a basic problem in morphological biology (Liick & Liick, 1982). Much experimental and theoretical effort has been expended in attempts to describe and simulate the static forms o f tree structures with the use o f various morphological models (Liick & LiJck, 1982; Schkata, 1977; Shinozaki et al., 1964). On the other hand, the dynamical description of the developments of tree structure has not been studied extensively, and this is an important recent problem of pattern formation in non-equilibrium open systems in various scientific fields (Haken, 1979). For the dynamical description, the concept o f bifurcation ability o f branches and the stochastic property of bifurcation rate o f branches will play important roles. In fact, the ditterence in the bifurcation ability of terminal branches was introduced for the computer simulation of the form of Terminalia Cattappa L. (Honda, Tomlinson & Fisher, 1981), and the probabilistic L-system (KOL System) has been introduced in order to simulate almost, but not exactly, the same development o f the apical part of Japanese Cypress (Nishida, 1980). 667 0022-5193/85/040667+ 10 $03.00/0

O 1985 Academic Press Inc. (London) Ltd

668

M. AGU

AND

Y. YOKOI

In the present report, a systematic method is presented to describe the temporal growth of trees. The age of progeny branch is defined by the number o f branching (bifurcation) times occurring in the development process from an ancestor branch to the progeny branch. The development of the branching structure is described as the temporal change of the population nk(tl) of the branches of the age k at the time tt. The apical dominance of branching processes is introduced as the difference in the activity of the bifurcation of the branch. The stochastic description of the branching process is also given by introducing random bifurcation rates. The computer simulations of the stochastic description are carried out for a few examples to compare with the observed age distribution of real trees such as Celtis sinensis Ter. var. Japonica, Zelkova serrata Makino or Cedrus deodora Loud.

2. Description of Branching Structure Consider a branch system (tree structure) which is constructed from one ancestor branch and its progeny branches. The progeny branch is the branch born from the ancestor branch through successive bifurcations. The time course is treated as the discrete step, t~, I = 0, 1, 2 . . . . and in the course of time, the population o f progeny branches is assumed to increase as a result of bifurcation. Now we define the "age" of progeny branch by the number of the branching processes (or bifurcations) which occurred in the development process from an original ancestor branch to the progeny branch. The age of the ancestor branch is defined as zero at the time to = 0. The age o f each branch is uniquely determined if tree structure is specified. An example o f the age of the branch is shown in Fig. 1.

4

4

3

3 3

4 4 4 44 5 5(Age) Progeny branch

Ancestor branch

FIG. 1. Definition of branch age.

BRANCHING

S T R U C T U R E S OF TREES

669

(A) DETERMINISTIC DESCRIPTION

First, we study the deterministic description of branching patterns. In the course of time, the population of progeny branches is assumed to increase following the deterministic law such as dichotomous branching, trichotomous or Fibonacci branching, which are schematically shown in Fig. 2. Here our main interests are in the branching patterns, so that the

Dichotomous branching

Trichotomous branching

Fibonacci branching

FIG. 2. Typical examples of branching patterns.

branching angle or branch length is not taken into account in the present paper. The dichotomous or trichotomous branching structure is found in plant trees with relatively small height such as Viscum album, Edgeworthia papyrifera and so on. On the other hand, Fibonacci branching is found in relatively large erect trees such as Zelkova serrata Makino, Celtis sinensis, and so on. The term Fibonacci structure originates from the fact that the total number of the progeny branches increases following the Fibonacci number series 1, 2, 3, 5, 8, 1 3 , . . . , or its modified number series in the course of time. The problem considered hereafter is to derive the difference equation which describes the time course of the development of the branching system. The description of the age construction of the dichotomous branching structure is very simple. Denote the population of the branches of the age k at the time h by nk(h). This gives the difference equation in the case of dichotomous branching

nk( tt) = 2nk_l( tl_l).

(1)

Similarly, for the trichotomous branching structure

nk(h) = 3nk-l(h-I).

(2)

NOW we proceed to the important case. Taking into account the fact that apical dominance p h e n o m e n o n appear in many plant growth processes, we

670

M. AGU

Y. Y O K O I

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introduce two kinds of branches, apical branch (active branch) and subsidiary branch (inactive). The apical branch is assumed to have the activity of bifurcation, so that they bifurcate at each time step, giving rise to one apical branch and another subsidiary one. On the other hand, the subsidiary branch does not bifurcate but changes its inactive property to the active type at each time step. Denote the population of the apical (or subsidiary) branches of the age k at the time tl by n~+~(h) (or n(k->(h)). Then the above considerations are summarized in the following coupled difference equations (+)

(3)

nk+l( h+,) -- n~-+),(tt)+ n~+~(h) (-)

nk+,(h+,) = n~+~(h)-

(4)

The total number of the branches at the time h is given by

n( h) = ~ ( n~+~(h)+ n~-)( h))

(5)

k

It is shown that equations (3), (4) and (5) give rise to the Fibonacci number series, under the initial conditions n~o+)(to)= 1 and n~+~(to)= n~-)(to) for finite k.

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FIG. 3. Age construction rate of Fibonacci branching.

BRANCHING

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671

n ~ ( t , ) = n~÷~(t,) + n V ~ ( t , ) =

=

nk-i(t,-t)

-- "k-I (tt-l)

n?2~(t,_~)+ . r-~÷~'" a k - l \ ' l - ! )+ n?2,(t,_:)+,, ~÷'~_2(t,_2)

= nk-l(h-t)+ nk-i(h-:).

(6)

Then, with the use of equation (6), we obtain the recurrence formula, which is already well known as the formula of the Fibonacci number (Hoggatt, 1969; Stevens, 1974) n( h) = n( t,-i) + n(t,_2) (7) with the initial conditions, n(to) = n(h) = 1. Thus it was found that the Fibonacci branching in which the population of terminal branches increases following equation (7) is a direct consequence of the apical dominance phenomenon. The successive use of the recurrence formula (6) gives rise to the explicit expression for nk(h). nk( h) = nk-i( h-2) + nk-i( h-t) = nk-2(tl-2)

-1- 2 nk-2(It-3) q- I l k - 2 ( i t - 4 )

k-2

= E

n2(tl-k+2-3,)k-2Cr

r=O

= k-2CI-k-3 + 3k-2Cl-k-I + 2k-2Ct-k -

(k-1)!(3k-l) (2k-l)!(l-k)!"

(8)

The age construction rate is obtained with the use of equation (8)

nk(h) ~ (k-l)!(3k-l) xk(h) = n(t,) = k=L/2] ('2-/cS-'~ i (--1---"~'[

(9)

where [I/2] is an integer given by I/2 for even l and by (I+ 1)/2 for odd I. The numerical example of the temporal evolution of Xk (tl) is given schematically in Fig. 3, which shows that the age construction rate behaves like unidirectional diffusion in age space. The maximum value among the ages of the progeny branches at time t~ coincides with the value of the time step tl itself. As a general case, we introduce the various levels of activity in bifurcation with the use of the signs ( + ), ( - ), ( - - ) , ( - - - ) . The number of the levels of activity i s determined by the strength of the apical dominance phenomenon: the stronger the apical dominance is the larger the number of the levels. The population of the branches of the age k at the time h with

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M. A G U A N D Y. Y O K O I

the activity ( . . . . . . ) is denoted by n(k- . . . . . ~. Then the evolution equations (3) and (4) are modified into the forms 1(k+l(tt+,) = n(k+~l(t,) + n(k+)(tt) (-)

nk+l( t,+l) = n~k-+;~(t,) It(k-+7 .... )( tt+l ) = It(k+)( tt).

(3') (4') (4")

Here the active branch ( + ) is determined as bifurcating at each time step, giving rise to one apical b r a n c h ( + ) and another subsidiary branch with the least activity ( . . . . . . ). The activity of the subsidiary branch increases step by step as the time step goes on until it becomes apical branch with the activity ( + ) . The branching process with the three levels, ( + ) , ( - ) , ( - - ) , of activity is shown in Fig. 4. Here the plus ( + ) , minus ( - ) or minus-minus ( - - ) sign attached at the top of each branch indicates the activity of each branch. (÷)

(-.Z,~

/

C-)

FIG. 4. Tree structure with apical dominance. The signs ( + ) , ( - ) , ( - - ) show the activity of bifurcation of each branch.

Using the similar arguments to the case of the previous Fibonacci branching, we have, instead of equation (6), nk( tt) = nk-t( t*-i) + nk-t( tt-s)

(10)

where the subscript s stands for the number of levels of activity of bifurcation process. Equation (1 t ) defines a type of generalized Fibonacci number satisfying the recurrence formula n( tl) = n( tt-l) + n( ti-s).

(11)

The explicit expression of nk(t~) is obtained from equation (11) in the form nk( tt) = ~. kf~nl( tt--k+l - V ( S - 1)) v

(12)

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where the summation with respect to v is carded out over all integers satisfying s < - l - k + l - v ( s - l ) < = 2 s - I and the total number nl of the branches of the age one, is equal to two at the time t = ts and equal to one at all other times t <=t2s-t. (B) S T O C H A S T I C D E S C R I P T I O N

So far, we have shown the deterministic or ideal description of typical branching structures. However, real tree structure does not always develop following deterministic law. A possible reason for the stochastic property of branching might be attributed to internal fluctuations occurring in the concentrations of the chemical substances which promote or inhibit the bifurcations, or to the fluctuations of external environments. As a method for taking these into account, the branching rate yc+l+.-)(h) at which the active branch bifurcates and the changing rate yc-I+)(h) at which the inactive branch changes into the active one are introduced into the Fibonacci branching. Then equations (3) and (4) are modified into the forms: (+)

nk+l(h+,) -- Y(-l+)(h)n~k+)l(h)+ y{+i+.-}(h)nck+)(h) + ( 1 - yc+l+. -)(h)) n ek+),(tt) (-)

nk+l(h+i) = Y~+l+'-)(h)nCk+)(h)+(l yc-I+)(h))nck-+)l(h).

(13) (14)

Here if we choose y<+l+,-) and yc-I+) as unity, then equations (11) and (12) reduce to equations (3) and (4). In general these transition rates To+l+.-) and y<-I+) are stochastic variables, then equations (6) and (7) become stochastic difference equations with randomly varying coefficients. Similarly, equations (3)-(4') are rewritten as c+) h+t) = yc-t+)( h)n~-+)t(h) + ~/(+l+'- ..... )( tt)r~k( h) nk+,( + (1 - y(+t+, - . . . . . )(t,))n~>l(h)

(15)

(-) nk+l(tt+t)

(16)

= TC-

- I

-)(h)nC-+~)(h)+(l-yC-I+)(tt))nCZ+)l(tt).

Finally we note the fact that the branch of the natural tree may wither in terms of the change of various external environments. If the inactive branch ( - ) withers with the rate w-(h), equation (14) is rewritten as

nk+l(h+z) (-) = y(+l+ -~(h)n~k+)(h)+(l - y~-I+)(tt)- wC-)(h))n~-+)l(tt) (17) 3. Simulations of Actual Branching Systems Let us now examine how the stochastic description of trees in section 2 applies to the actual structures. Three types of tree varying in "sharpness"

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of shape were observed. Sharpness is considered to correspond to the strength o f the apical dominance. In the first case, a branch of Celtis sinensis o f diameter 33-45 mm having 663 progeny branches was studied--this is the least sharp of the trees examined. The measured age construction rate is shown in Fig. 5. The computer simulations were made by assuming the 20

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0

5

tO Age of bronch

15

20

FIG. 5. Age construction rate of a branch of Celtis sinensis. Here dots show the observed data and crosses the simulated data in which (3,~+1÷"-)> = 0.5 and (y(-~+~>= 0.2 are assumed.

Fibonacci branching described by equations (13) and (14) having the random bifurcation rates with the averages: (3, ~÷1+"->> = 0-5, (3, C-I÷~>= 0-2. These average values are determined by a 'cut-and-try' method so as to adjust the total number o f branches and the age construction rate to the observed values. Figure 5 shows that the simulated results in relatively good agreement with those of the measured age construction rate. In the second case, a branch of Zelkova serrata Makino o f diameter 11 mm having 183 progeny branches was examined. The observed data on the age construction rate of the branch is shown in Fig. 6. The stochastic tree having three activity levels with random bifurcation rates was simulated, in which the averages of the bifurcation rates were assumed as (3, C÷l÷'- -)> = (y~-I+)> = 0.7, (y~- - I - ) ) = 0.2. The deterministic tree could not be adjusted to the observed one both in the age construction rate and in the total number of branches. For the third case, a branch of Cedrus deodara Loud. o f diameter 16.6 mm having 217 progeny branches is also examined--this is the sharpest of the trees examined. The age construction rate is shown in Fig. 7. The computer simulation o f this branch is carried out assuming modified Fibonacci branching with four activity levels, ( + ), ( - ), ( - - ) , ( - - - ) . The random bifurcation rate o f each level is assumed to have the averages: (y~+l+, - - -)) = 1.0, (y(-i+)) = 0.2, (y(- -J -~> =0.2, (y(- - -T - -~) = 0.2.

BRANCHING

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Age of branch FIG. 6. Age construction rate of a branch of Zelkova serrata Makino. Here dots show the observed data, crosses the simulated data in which ( y ( + l + ' - - ) ) = 0 . 7 , (y(-~+>)=0.7 and (y(--I-))=0'2 are assumed and triangles the simulated data of the deterministic case, y(+l +, - -) = y(-l+)

= y(-

- I -) = I '0.

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Age of branch FIG~ 7. A g e c o n s t r u c t i o n r a t e o f a b r a n c h o f Cedrus deodara L o u d . H e r e d o t s s h o w t h e o b s e r v e d d a t a , c r o s s e s t h e s i m u l a t e d d a t a in w h i c h ( y ( + l + ' - - - ) ) = l . 0 , (y(-1+))=0.2, ( y ( - - t-) ) = 0 - 2 a n d ( y ( . . . . l ) ) = 0 - 2 a n d t r i a n g l e s t h e s i m u l a t e d d a t a o f t h e d e t e r m i n i s t i c c a s e , y(+l+, - - - ) = y ( - t + ) = 7 ( - - - i - ) = y ( - - - ! - - ) = 1.0.

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M. A G U A N D Y. Y O K O I

The results of the computer simulations are in fairly good agreement with those of the observed data. The deterministic tree could not be adjusted to the observed example as in the previous case. We note from the above simulations that the number of activity levels of the simulated trees increases in proportion to the sharpness of tree shape, i.e. the strength of the apical dominance. The agreement between the observed data and the simulated data show that the present stochastic formulation is valid for systematic descriptions of various tree structures. The formulations will be widely applicable to other kinds of trees by choosing the appropriate number of the activity levels of bifurcations and the stochastic property of the bifurcation rates. Fluctuations of external environments can also be taken into account by stochastic bifurcation rates. In order for the stochastic property of the bifurcation rates to be described more clearly, the structural fluctuations inherent to tree itself must be shown to be separate from the fluctuations caused by the external environments. Finally we note that the present formulation will also be applicable to the description of other kinds of tree structure such as cracks in solid materials or discharge paths in electrical breakdown. The conclusions of the present paper are summarized as follows: (1) The temporal growth of trees was described as the change of the age construction rate of the branch. (2) The apical dominance p h e n o m e n o n was introduced as the difference in the bifurcation ability of the branch. Fibonacci branching was found to be a direct consequence of the apical dominance. (3) The age construction rates of the trees, Celtis sinensis Pers, var. Japonica, Zelkova serrata Makino and Cedrus deodara Loud. were examined and found to be expressed well by the present stochastic formulation. REFERENCES HAKEN, H. (ed), (1979). Pattern Formation by Dynamic System and Pattern Recognition. Berlin: Springer-Verlag. HOGGATr, V. E. JR. (1969). Fibonacci and Lucas Number. Boston: Houghton Mifflin Company. HONDA, H., TOMLINSON, P. & FISHER, J. B. (1980). Am. J. Bot. 68, 569. LOCK, J. & LOCK, H. B. (1982). Ber. Deutsch. Bot. Ges. Bd 95, 75 (and the references therein). NisrtIDA, T. (1980). Memories of the Faculty of Science. Kyoto Univ. Series of Biology, VoI v l l l , no. 1, 97. SCHKATA, M. (1977). J. Humanit. Nat. Sci. (Tokyo College of Economics) 47, 69. SHINOZAKI, K., YODA, K., HOZUKI, K, & KtRA, T. (1981)..lap. J. Ecol. 14(3), 97. STEVENS, P. S. (1974). Patterns in Nature. Boston: Little, Brown and Company.