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ORIGINAL PAPER
Deciphering the ontogeny of a sympodial tree
Evelyne Costes • Yann Guedon
Received: 26 April 2011 / Revised: 18 August 2011 / Accepted: 23 November 2011 / Published online: 13 December 2011
� Springer-Verlag 2011
Abstract This paper addresses the identification and
characterization of developmental patterns in the whole
structure of a sympodial species, the apple tree. Dedicated
stochastic models (hidden variable-order Markov chains)
were used to (i) categorise growth units (GUs) on the basis
of their morphological characteristics (number of nodes
and presence/absence of flowering) and position along
axes, (ii) analyse dependencies between successive GUs
and (iii) identify repeated patterns in GU sequences. Two
successive phases, referred to as ‘‘adolescent’’ and ‘‘adult’’,
were identified in two apple tree cultivars. In the adolescent
phase, ‘‘very’’ long monocyclic GUs were followed by
long polycyclic GUs, whereas in the adult phase medium
GUs were preferentially followed by short GUs. Flowering
GUs constituted a preferential pathway between vegetative
GUs of decreasing vigour (long, medium and short) and
generated patterns that were interpreted with respect to
fruiting regularity. The proposed modelling gave a global
and quantitative picture of the two-scale structuring of
apple tree ontogeny: a coarse scale corresponding to the
succession of the previously mentioned phases and a fine
scale corresponding to the alternation between flowering
and vegetative GUs. This led us to propose a synthetic
scheme of apple tree ontogeny that combines growth
phases, polycyclism and flowering, and which could be
transposed to other sympodial trees.
Keywords Floral differentiation � Growth phase � Hidden
Markov model � Tree architecture � Variable-order Markov
chain
Introduction
Plant development results from meristem activity occurring
through a sequence of developmental phases, abstracted by
the term ‘‘ontogeny’’. During ontogeny, the morphological
characteristics of plant entities such as growth units (GUs)
or annual shoots change over time (Nozeran 1984). Such
changes have been investigated in a number of forest and
fruit tree species (e.g. Sabatier and Barthelemy 1999;
Suzuki 2002; Costes et al. 2003; Solar and Stampar 2006)
on the basis of sampling of shoot categories. The floral
differentiation of a meristem is a key developmental stage
in plant ontogeny. Contrary to monocarpic plants where
floral differentiation occurs in all aerial meristems and ends
the plant’s life cycle, floral differentiation in perennial
polycarpic species occurs recurrently throughout ontogeny
but in a proportion of meristems only (Bangerth 2009). The
occurrence of the first flowering ends up the juvenile phase,
the plant then entering its mature phase (Hackett 1985).
During the mature phase, the floral differentiation of shoot
meristems is not ‘‘automatic’’, but has often been reported
as irregular, in particular in fruit trees; see Wilkie et al.
(2008) and references therein. After floral differentiation of
a shoot apical meristem, growth resumes usually by sym-
podial branching, i.e. through activation of one or several
lateral meristems (Halle et al. 1978). The development of a
Communicated by A. Braeuning.
E. Costes
INRA, UMR AGAP, Architecture and Functioning of Fruit
Species Team, 34398 Montpellier, France
e-mail: [email protected]
Y. Guedon (&)
CIRAD, UMR AGAP and INRIA, Virtual Plants,
34398 Montpellier, France
e-mail: [email protected]
123
Trees (2012) 26:865–879
DOI 10.1007/s00468-011-0661-8
polycarpic sympodial plant is thus characterized, at a fine
scale, by the occurrence of repeated patterns (also called
words or motifs) resulting from the alternation between
flowering and vegetative growth and, at a coarser scale, by
two successive developmental phases, the juvenile and
mature phases. This two-scale structuring can be consid-
ered as a general property of polycarpic sympodial plants
even though the growth characteristics at each scale, such
as the length of a phase or the number of nodes of GUs, are
modulated by the environmental conditions. The apple tree
is an interesting example of a polycarpic sympodial tree
since it combines terminal flowering followed by sympo-
dial branching (Crabbe and Escobedo-Alvarez 1991) with
the capability to develop polycyclic shoots (i.e. shoots that
include several GUs within a year, Halle et al. 1978). Its
architectural development has been largely studied, but
despite several studies dedicated to the apple tree estab-
lishment phase (Costes et al. 2003) or to the maintenance
of equilibrium between fruit and leaf production in the
mature phase (Lauri et al. 1995), there is no global and
quantitative picture that integrates these two scales of
structuring of apple tree ontogeny. This is mainly due to
the irregularity of flowering combined with the complexity
of perennial plant structures. In the present study, we aimed
at proposing a new modelling approach to capture simul-
taneously the two scales at which polycarpic sympodial
trees organise during ontogeny, taking the apple tree as a
case study. In particular, we addressed two main questions:
(i) how do the patterns of alternation between flowering
and vegetative GUs change between phases during tree
ontogeny? (ii) Are there useful markers of transition
between phases?
We propose to apply models combining structure with
probabilities for analysing complex developmental patterns
often found in perennial plants. Probabilities enable to
represent fluctuations due to unobservable biological
functions and their interactions acting at more microscopic
scales than the scale of measurement. Generally, the
objective of this type of analysis is to identify and char-
acterise complex patterns which cannot be grasped directly
from the data rather than to quantify well-known behav-
iours. In this respect, the analysis of complex perennial
plant patterns fits into the general framework of pattern
theory (Grenander and Miller 2006) and the models of
interest are often stochastic models. Different stochastic
models have previously been proposed to analyse tree
structure development at the GU or annual shoot scale
(Durand et al. 2005; Guedon et al. 2007; Chaubert-Pereira
et al. 2009). In these studies, it was assumed that the
functioning of a meristem during a growing period influ-
ences its functioning during subsequent periods. As in
these previous studies, we assumed dependencies between
successive GUs built by the same meristem or deriving
from one another by sympodial branching. In Durand et al.
(2005), it was assumed that the differentiation state of
meristems can be deduced from the morphological char-
acteristics of the GUs, in particular the number of nodes,
the length and the presence/absence of flowering. A sto-
chastic model, called the hidden Markov tree model, was
applied to infer the differentiation state sequences in two
tree species (apple tree and bush willow). In apple tree,
four GU types were identified: three vegetative (long,
medium and short) and one flowering, and these were
shown to have specific locations in trees. However, the
hidden Markov tree model did allow neither to model the
dependency of a GU beyond its parent GU nor to highlight
successive phases. Here, we again chose to apply a ‘‘hid-
den’’ model (since GU types will be deduced from GU
morphological characteristics) to analyse frequent patterns
that include more than two successive GUs, for instance
three GUs, e.g. [flowering GU, vegetative (either long
medium or short) GU, flowering GU]. Most of the methods
for analysing local dependencies in sequences rely on high-
order Markov chains. However, the number of free
parameters of a Markov chain increases exponentially with
its order, i.e. with the memory length taken into account.
For instance, in the case of four states (corresponding to
four GU types), the number of free parameters is 3 for a
zero-order, 12 for a first-order, 48 for a second-order
Markov chain, etc. Since there are no models ‘‘in
between’’, this very discontinuous increase in the number
of free parameters causes the estimated high-order Markov
chains to be generally overparameterized. This drawback
can be overcome by defining sub-classes of parsimonious
high-order Markov chains such as variable-order Markov
chains (Ron et al. 1996; Buhlmann and Wyner 1999) where
the order is variable and depends on the ‘‘context’’ within
the sequence, instead of being fixed. We thus chose to
model the sequence of meristem differentiation states
during apple tree ontogeny using a hidden variable-order
Markov chain, the objective being to draw a global picture
of the ontogeny of a perennial sympodial species. Our goal
was to characterise jointly successive growth phases,
changes in GU morphological characteristics and devel-
opmental events such as terminal flowering.
Materials and methods
Plant material and tree description
The studied trees belong to two apple scion-cultivars,
‘Braeburn’ and ‘Fuji’ that were grafted on Lancep Pajam�1
(type M9). Trees were planted with 1.8 9 6 m spacing in
the winter of 1994–1995 at the Melgueil INRA experi-
mental station (South East France). In the first year of
866 Trees (2012) 26:865–879
123
growth the trees were trained in a vertical axis by selecting
long axillary shoots along the main axis which was main-
tained without pruning. In the spring of the third year, the
trunks and long shoots were bent. In June of years 4–6,
young fruits were thinned manually to one fruit per inflo-
rescence and all axillary fruits were removed on 1-year-old
wood.
Two 6-year-old apple trees for each of the Braeburn and
Fuji cultivars were fully described; see Costes et al. (2003).
Each tree was described using three scales of organisation
corresponding to the axes, GUs and metamers (as defined
by White 1979). Flowering GUs for which the apical
meristem had differentiated into an inflorescence were
distinguished from vegetative GUs (Fig. 1). The number of
nodes was counted on the extension GUs only (for non-
extended vegetative GUs referred to as short GUs in the
following, the number of nodes was set at the default value
of 1). Bud scars and metamers whose axillary buds were
not visible to the naked eye were ignored.
GU sequence modelling
From the initial database, sequences of GUs were extracted
for all axes, including the trunks (Fig. 1). Each sequence
thus corresponded to a new lateral branching (except the
main axes of the trees). After a flowering occurrence, the
new axis arising from sympodial branching was considered
as the continuation in the sequence. In apple tree, this new
axis is called ‘‘bourse shoot’’. When two axes arose from
the same flowering GU, the most distal one was chosen as
the continuation. When no continuation was observed, the
sequence was completed by a final ‘‘dead’’ GU, according
to the phenomenon called ‘‘extinction’’ by Lauri et al.
(1995). Each sequence included three variables: year of
growth, number of nodes and non-flowering/flowering
character of the GU. In all, 1,194 sequences of cumulated
length 5,523 were extracted from the two Braeburn trees
and 2,034 sequences of cumulated length 6,072 were
extracted from the two Fuji trees. For a given cultivar, the
cumulated length of the sequences is the total number of
GUs of the two trees.
In the following, we first introduce high-order Markov
chains. Then, variable-order Markov chains and hidden
Markov models based on variable-order Markov chains,
which are the stochastic models used in this study, are
defined. In the case of an rth-order Markov chain {St;
t = 0, 1, …}, the conditional distribution of St given
S0, …, St-1 depends only on the values of St-r, …, St-1
but not further on S0, …, St-r-1,
P St ¼ stjSt�1 ¼ st�1; . . .; S0 ¼ s0ð Þ¼ P St ¼ stjSt�1 ¼ st�1; . . .; St�r ¼ st�rð Þ:
In our context, the random variable St represents GU
type and can take the four possible values L, M, S and F, for
long, medium, short and flowering, respectively. These
four possible values correspond to the Markov chain states.
A supplementary ‘‘end’’ state is added to formalise axis
death (denoted by D in the following). A J-state rth-order
Markov chain has Jr(J - 1) independent transition
probabilities. Therefore, the number of free parameters of
a Markov chain increases exponentially with the order. Let
the transition probabilities of a second-order Markov chain
be given by
phij ¼ P St ¼ jjSt�1 ¼ i; St�2 ¼ hð Þ withX
j
phij ¼ 1:
These transition probabilities can be arranged as a J2 9 J
matrix where the row (phi0,…,phiJ-1) corresponds to the
transition distribution attached to the [state h, state i]
memory. If for a given state i and for all pairs of states
(h, h0) with h 6¼ h0, phij ¼ ph0ij, i.e. once St-1 is known, St-2
conveys no further information about St, the J memories of
length 2 [state h, state i] with h = 0, …, J - 1 can be
grouped together and replaced by the single [state i]
memory of length 1 with associated transition distribution
(pi0, …, piJ-1). This illustrates the principle for building a
variable-order Markov chain. In a variable-order Markov
chain, the order (or memory length) is variable and depends
on the ‘‘context’’ within the sequence. The memories of a
Markov chain can be arranged as a memory tree such that
each vertex (i.e. element of a tree graph) is either a terminal
vertex or has exactly J ‘‘offspring’’ vertices. In practice, the
memories corresponding to unobserved contexts are not
included in the memory tree (this is the case for the memory
FF that was not observed in our conditions). The memories
associated with the J vertices (memories of length r ? 1)
deriving from a given vertex (memory of length r) are
Fig. 1 Schematic representation of a 4-year-old apple tree branch at
the growth unit (GU) scale. GUs of each axis are labelled according to
their floral (F) or vegetative nature, and, in this latter case, according
to their number of nodes (L for long, M for medium and S for short).
The GU sequence along the branch is LLFLLFLFL. Four examples of
GU sequences are provided for lateral shoots: SSFS, LFMFM, SFSand MFM
Trees (2012) 26:865–879 867
123
obtained by prefixing the parent memory with each possible
state. For instance, in Figs. 2a and 3a, the 3 second-order
memories LF, MF and SF derive from the first-order
memory F. A transition distribution is associated with each
terminal vertex of this memory tree.
A specific problem with variable-order Markov chains
concerns the definition of initial probabilities for deter-
mining the first state in the sequence. We adopted the
approach proposed by Ron et al. (1996). They showed that
the model (consisting of the ‘‘permanent’’ memories cor-
responding to the terminal vertices of the memory tree) can
be augmented with the memories corresponding to the
interior vertices to model the beginning of the sequences.
These supplementary memories are called ‘‘transient’’.
They can only be visited once at the beginning of the
sequence while permanent memories can be visited more
than once in a sequence. With these supplementary tran-
sient memories, the initial probabilities pj = P(S0 = j) are
defined as usual for first-order memories. For instance, in
the case of the model shown in Fig. 2a, a sequence can be
initialized in the initial transient memory denoted by F0
(and not in the permanent memories LF, MF and SF
deriving from F). Since each sequence corresponds to a
new lateral branching (except the main axes of the trees),
the initial probabilities represent the branching probabili-
ties estimated globally for the entire trees. The building of
a variable-order Markov chain consists in selecting the
memories in an optimal way (Csiszar and Talata 2006) and
estimating the corresponding initial and transition proba-
bilities; see ‘‘Appendix A’’ for the principle of the selection
of the memories.
In this study, instead of first classifying GUs in cat-
egories and then characterising the succession of GU
types along the axes, we rather chose to merge these two
steps and build a single integrative statistical model that
can be optimally estimated. In this statistical model,
which is a hidden Markov model (Ephraim and Merhav
2002) based on a variable-order Markov chain, the non-
observable variable-order Markov chain represents the
succession of GU types along the axes. The GU types
are not observable directly but only indirectly through
the two observed variables, namely the number of nodes
and the non-flowering/flowering character of the GU. A
hidden variable-order Markov chain can be viewed as a
two-level stochastic process, i.e. a pair of stochastic
processes {St, Xt} where the ‘‘output’’ process {Xt} is
related to the ‘‘state’’ process {St}, which is a finite-state
variable-order Markov chain, by the observation
probabilities
bj yð Þ ¼ P Xt ¼ yjSt ¼ jð Þ withX
y
bj yð Þ ¼ 1:
The definition of the observation probabilities expresses
the assumption that the output process at time t depends
only on the non-observable Markov chain at time
t. Extension to the multivariate case is straightforward
since, in this latter case, the observed variables at time t are
assumed to be conditionally independent given the state
St = st. The probability of observing a vector is thus
simply the product of the probabilities of observing each
variable. There is indeed no observation distribution
attached to the death ‘‘end’’ state.
In the case of a small set of possible outputs (as in the
case of a binary observed variable such as non-flowering/
flowering), the observation probabilities are estimated
directly and can be arranged as a J 9 N matrix (N is the
number of possible outputs) with all rows summing to one.
In the case of a larger set of possible outputs (assumed to
be generated by a count variable such as the number of
nodes of a GU), discrete parametric observation distribu-
tions are estimated. In this study we chose to use as pos-
sible discrete parametric observation distributions,
binomial distributions, Poisson distributions and negative
binomial distributions with an additional shift parameter;
Fig. 2 Variable-order Markov chain for Braeburn cultivar. L, M, S, F,
stand for long, medium, short and flowering, respectively. a Memory
tree: the memory tree must be read from the left to the right, and each
‘‘column’’ corresponds to memories of a given length; b Transition
graph: each vertex represents a possible memory (of length 1 or 2).
The vertex edged by a dashed line corresponds to the initial transient
memory (visited once at the beginning of a sequence) while the
vertices edged by a continuous line correspond to permanent
memories (visited more than once in a sequence). Transitions with
associated probability [0.04 are represented by arcs. The associated
probabilities are noted nearby. Transitions towards the death end state
are not shown
868 Trees (2012) 26:865–879
123
see ‘‘Appendix B’’ for formal definitions of these
distributions.
The maximum likelihood estimation of the parameters
of a hidden variable-order Markov chain requires an iter-
ative optimisation technique, which is an application of the
Expectation–Maximization (EM) algorithm (Ephraim and
Merhav 2002). Once a hidden variable-order Markov chain
has been estimated, the most probable state sequence can
be computed for each observed sequence using the so-
called Viterbi algorithm. This state sequence, which is
called the restored state sequence, can be interpreted as the
optimal labelling of the observed sequence where a type
chosen from among long, medium, short and flowering is
affected to each successive GU.
Different methods have been proposed to assess the
modelling of repeated patterns by (hidden) variable-order
Markov chains (Guedon et al. 2001). Here, we will focus
on recurrence time distributions which appear to be par-
ticularly appropriate for analysing flowering regularity
over time. The recurrence time in a given state is defined as
the number of transitions between two occurrences of this
state. Recurrence time distributions can in particular help
to highlight pseudo-periodicities in the successive occur-
rences of a given state along sequences.
All the statistical analyses were made using the statis-
tical package of VPlants software integrated in the
OpenAlea platform and available at http://openalea.gforge.
inria.fr/wiki/doku.php?id=openalea.
Results
Selection of the memories of the variable-order Markov
chains
In a first step, a four-state hidden first-order Markov chain
was estimated for each cultivar on the basis of the sequences
of GUs where the two observed variables were the number of
nodes and the non-flowering/flowering character. The four
states of the underlying first-order Markov chain corre-
sponded to long, medium, short and flowering GUs; see
below for the characterization of these states in the case of
hidden variable-order Markov chains. This is a direct
transposition to sequences of the modelling approach pro-
posed by Durand et al. (2005) in the case of tree-structured
data. The restored state sequences were then computed using
the estimated hidden first-order Markov chain and the
memories of a variable-order Markov chain were selected on
the basis of these restored state sequences; see ‘‘Appendix
A’’ for the principle of the selection of these memories. This
approach is justified by the fact that (i) the labelling of the
observed sequences was unambiguous for sequences which
Fig. 3 Variable-order Markov
chain for Fuji cultivar. L, M, S,
F, stand for long, medium, short
and flowering, respectively.
a Memory tree: the memory tree
must be read from the left to the
right, and each ‘‘column’’
corresponds to memories of a
given length; b Transition
graph: each vertex represents a
possible memory (of length 1 or
2). The vertices edged by a
dashed line correspond to the
initial transient memories
(visited once at the beginning of
a sequence) while the vertices
edged by a continuous linecorrespond to permanent
memories (visited more than
once in a sequence). Transitions
with associated probability
[0.04 are represented by arcs.
The associated probabilities are
noted nearby. Transitions
towards the death end state are
not shown. The dotted arcscorrespond to infrequent
transitions and the vertex edged
by a dotted line corresponds to
an infrequent permanent
memory
Trees (2012) 26:865–879 869
123
do not contain extension GUs, and (ii) the posterior proba-
bilities of the restored state sequences (i.e. weight of the
restored state sequence among all the possible state
sequences that can explain a given observed sequence) were
most often high for the observed sequences which contain
extension GUs; see below further comments in the case of
hidden variable-order Markov chains. For both cultivars, the
variable-order Markov chains were mixed first-/second-
order Markov chains. For the Braeburn cultivar, the memo-
ries of the variable-order Markov chain (i.e. terminal vertices
of the memory tree) were {L, M, S, LF, MF, SF} while for the
Fuji cultivar, the memories were {L, M, LS, MS, SS, FS, LF,
MF,SF}; see the memory tree for Braeburn in Fig. 2a and for
Fuji in Fig. 3a. Thus, for both cultivars, the type of the GU
following a long or medium ‘‘parent’’ GU did not depend on
the type of the ‘‘grand-parent’’ GU (first-order memories L
and M) while the type of the GU following a flowering
‘‘parent’’ GU depended on the type of the ‘‘grand-parent’’
GU (second-order memories LF, MF, SF deriving from F).
The difference between the two cultivars concerned the type
of the GU following a short ‘‘parent’’ GU: this depended on
the type of the ‘‘grand-parent’’ GU for Fuji (second-order
memories LS, MS, SS, FS deriving from S), but not for
Braeburn (first-order memory S).
Estimation of hidden variable-order Markov chains
In a second step, a hidden variable-order Markov chain was
estimated for each cultivar where the underlying variable-
order Markov chain had the memories previously selected,
i.e. {L, M, S, LF, MF, SF} for Braeburn and {L, M, LS, MS,
SS, FS, LF, MF, SF} for Fuji. The iterative maximum
likelihood estimation procedure (EM algorithm) was ini-
tialized for the underlying variable-order Markov chain
such that all the transitions were possible. Specific transi-
tion probabilities were estimated for the initial transient
memories (F0 for both cultivars and S0 for Fuji) because of
the large number of short sequences in these data samples;
see the corresponding counts in Table 1. More generally,
the high values for most of the transition counts give
empirical evidence of the accuracy of the estimated tran-
sition probabilities. The estimated observation probability
matrix for the non-flowering/flowering variable was
degenerate (i.e. the estimated observation probabilities
Table 1 Transition probabilities of hidden variable-order Markov
chains estimated for the Braeburn and Fuji cultivars (memories in
rows, states in columns); e.g. the probability in the cell (long
flowering, medium) is pLFM ¼ P St ¼ MjSt�1 ¼ F; St�2 ¼ Lð Þ where L,
M, S, F stand for long, medium, short and flowering, respectively
Transition probability (next memory)
Long Medium Short Flowering Death Count
Braeburn memory
Long 0.3 (L) 0.03 (M) 0.04 (S) 0.61 (LF) 0.02 (D) 495
Medium 0 0.03 (M) 0 0.96 (MF) 0.01 (D) 574
Short 0.01 (L) 0.03 (M) 0.04 (S) 0.9 (SF) 0.02 (D) 1,238
Initial flowering 0.15 (L) 0.11 (M) 0.74 (S) 0 0 789
Long flowering 0.49 (L) 0.5 (M) 0 0 0.01 (D) 255
Medium flowering 0 0.27 (M) 0.72 (S) 0 0.01 (D) 348
Short flowering 0.04 (L) 0.12 (M) 0.83 (S) 0 0.01 (D) 683
Fuji memory
Long 0.31 (L) 0 0.01 (LS) 0.68 (LF) 0 444
Medium 0.05 (L) 0.06 (M) 0.05 (MS) 0.8 (MF) 0.04 (D) 494
Initial short 0.03 (L) 0.02 (M) 0.29 (SS) 0.66 (SF) 0 975
Long short 0.31 (L) 0 0 0.69 (SF) 0 2
Medium short 0 0 0.13 (SS) 0.74 (SF) 0.13 (D) 17
Short short 0.01 (L) 0.01 (M) 0.18 (SS) 0.6 (SF) 0.2 (D) 262
Flowering short 0.04 (L) 0.06 (M) 0.41 (SS) 0.38 (SF) 0.11 (D) 413
Initial flowering 0.05 (L) 0.14 (M) 0.81 (FS) 0 0 676
Long flowering 0.55 (L) 0.33 (M) 0.03 (FS) 0 0.09 (D) 195
Medium flowering 0.01 (L) 0.58 (M) 0.38 (FS) 0 0.03 (D) 158
Short flowering 0.03 (L) 0.13 (M) 0.79 (FS) 0 0.05 (D) 571
The memory reached by a transition is indicated between brackets after the corresponding probability. The transition counts (last column) were
extracted from the state sequences restored using the hidden variable-order Markov chains estimated for the Braeburn and Fuji cultivars. The
initial transient memories, which correspond to first states in a sequence, are indicated in italics
870 Trees (2012) 26:865–879
123
were either 0 or 1), with three vegetative states and one
flowering state (row: state L, M, S, F; column: non-flow-
ering/flowering):
B ¼
1 0
1 0
1 0
0 1
0BB@
1CCA:
This resulted from the iterative estimation procedure
which was initialized with a model where the observation
distributions for the non-flowering/flowering variable were
not degenerate.
The free parameters of the estimated hidden variable-
order Markov chains decompose in:
• Braeburn cultivar: 3 independent initial probabilities,
19 independent transition probabilities (Table 1),
(instead of 14 independent transitions probabilities in
the case of an underlying first-order Markov chain),
• Fuji cultivar: 3 independent initial probabilities, 31
independent transition probabilities (Table 1), (instead
of 13 independent transitions probabilities in the case of
an underlying first-order Markov chain);
• 11 free parameters for the observation distributions
estimated for the number of nodes variable (and no free
parameters for the observation distributions estimated
for the non-flowering/flowering variable).
The number of free parameters seems reasonable in view
of the sample sizes (cumulated length of the sequences:
5,523 for Braeburn and 6,072 for Fuji). It should be noted
that the death ‘‘end’’ state only adds one column in the
transition probability matrix and no observation distribu-
tions are defined for this state. The Bayesian information
criterion (BIC) favours the hidden variable-order chain
both for Braeburn (BIC12 = -20,075 instead of BIC1 =
-20,560 for a hidden first-order chain) and Fuji
(BIC12 = -22,783 instead of BIC1 = -23,470 for a hid-
den first-order chain); see ‘‘Appendix A’’ for a detailed
presentation of BIC in the case of variable-order Markov
chains. The rules of thumb of Jeffreys (see Kass and Raftery
1995) suggest that a difference of BIC of at least 2log
100 = 9.2 is needed to deem the model with the higher BIC
substantially better.
The graph of the possible transitions between memories
of the underlying estimated variable-order Markov chain is
shown in Fig. 2b for Braeburn and Fig. 3b for Fuji. In the
Braeburn model, a repetitive structure is apparent with
three groups of two memories: {L, LF}, {M, MF}, {S, SF}.
If we consider a vegetative state V which is L for the first
group, M for the second group and S for the third group, the
within-group transition probabilities pVF and pVFV are
always high ([0.49 except pMFM = 0.27) (Table 1). There
is also a preferential order of succession among the three
groups with transitions from the first towards the second
group and from the second towards the third group having
high probabilities: pLFM = 0.5 and pMFS = 0.72, respec-
tively. But the order of succession is not strict since it is
nevertheless possible to reach the second group from the
third (with pSFM = 0.12). The Fuji model (Fig. 3b) is
similar to the Braeburn model in terms of transition prob-
abilities except for the third group corresponding to state S.
The two-memory group {S, SF} of the Braeburn model is
replaced by a three-memory group {SS, FS, SF} in the
Fuji model. As seen previously, the within-group transition
probabilities pFSF, pSFS, pFSS and pSSF are all high
(Table 1). The difference between the cultivars is sup-
ported by counts for the second-order memories LS, MS,
SS and FS which are 2, 17, 262 and 413 for Fuji (see
Table 1) and 11, 0, 11 and 1,043 for Braeburn. Hence,
while for Fuji both the SS and FS memories are highly
represented in the data sample, only the FS memory is
highly represented for Braeburn, and the second-order FS
memory is therefore roughly equivalent to the first-order S
memory. Differences between the transition distributions
pLF0; . . .; pLFJ�1ð Þ; pMF0; . . .; pMFJ�1ð Þ and pSF0; . . .; pSFJ�1ð Þfor the second-order memories LF, MF and SF deriving
from F (and for the second-order memories MS, SS and FS
deriving from S in the case of Fuji) can be noted; see the
corresponding rows in Table 1. This is an a posteriori
justification of the selection of these second-order memo-
ries. For both cultivars, the distinction of the flowering
GUs on the basis of their vegetative ‘‘context’’ (with the
three memories LF, MF and SF) makes apparent the stages
in the succession of GUs. With a simple first-order Markov
chain where the three second-order memories LF, MF and
SF are collapsed onto a first-order memory F, these stages
cannot be identified since, whatever the type of the previ-
ous vegetative GU, vegetative GU of any type can followed
a flowering GU. As an illustration, among the patterns
VFV, only the patterns LFL, LFM, MFM, MFS and SFS can
occur with high probabilities in the variable-order Markov
chain case while all the patterns VFV including LFS, SFL
and SFM can occur with high probabilities in the first-order
Markov chain case.
It is noteworthy that the estimated hidden variable-order
Markov chains were only partially hidden since the vege-
tative and flowering GUs were differentiated unambigu-
ously by the non-flowering/flowering variable. Similarly,
short GUs were unambiguously defined by the number of
nodes variable set at the default value of 1. Hence, the
hidden character of the models only concerned the long and
medium GUs which were characterized by both their
number of nodes (see the corresponding observation dis-
tributions for states L and M in Fig. 4 and their character-
istics in Table 2) and position along the sequences (see
below). Consequently, the labelling of the observed
Trees (2012) 26:865–879 871
123
sequences was unambiguous for sequences which do not
contain extension GUs (609 out of 1,194 sequences for
Braeburn and 1,342 out of 2,034 for Fuji). For sequences
containing extension GUs, the posterior probabilities of the
restored state sequences were most often high: 48% above
0.8 and 91% above 0.5 for Braeburn on the basis of 585
sequences; 60% above 0.8 and 94% above 0.5 for Fuji on
the basis of 692 sequences. This justifies the use of
empirical distributions or characteristics deduced from the
restored state sequences for interpreting the output of the
estimated hidden variable-order Markov chains. It should
be noted that among the long GUs, there is a small
proportion of ‘‘very long’’ GUs which correspond to GUs
established during the first and second years of growth (18
for Braeburn and 13 for Fuji with more than 40 nodes
compared with a total of 533 and 512 long GUs, respec-
tively); see the tails of the corresponding frequency dis-
tributions in Fig. 4. Due to their small number, these very
long GUs could not be modelled as a supplementary state
in the hidden variable-order Markov chains. We checked
that the differences in mean number of nodes were small
and often statistically non-significant between sub-samples
of long GUs for years 3–6 and between sub-samples of
medium GUs for the different years (results not shown).
This is an a posteriori validation of the assumption of a
hidden Markov model based on a time-homogeneous
Markov chain. We also checked that the empirical number
of nodes distribution for extension GUs was well fitted by
the mixture of long and medium state observation distri-
butions (see Fig. 4a for Braeburn and Fig. 4b for Fuji)
using in particular P–P plots (plots not shown).
The accuracy of the estimated hidden variable-order
Markov chains for modelling patterns in the succession of
GUs can be illustrated by the improved fit of the recurrence
time distributions in the case of a hidden variable-order
Markov chain compared with a simple hidden first-order
Markov chain; see Fig. 5 for Braeburn and Fig. 6 for Fuji.
In these Figures, the empirical distributions were extracted
from the restored state sequences computed using the
estimated hidden variable-order Markov chain or hidden
first-order Markov chain (these empirical distributions only
differ for long and medium GUs between the two hidden
Markov models) while the theoretical distributions were
computed using the two compared hidden Markov chains;
see Guedon et al. (2001) for inclusion of the bias due to
short sequence length in the computation of the recurrence
time distributions.
Analysing model outputs: ontogenetic stages
For both cultivars, the transition probabilities show that a
vegetative GU (either long, medium or short) was prefer-
entially followed by a flowering GU (see the flowering
column in Table 1) while a flowering GU was systemati-
cally followed by a vegetative GU (this entails that the
estimated hidden variable-order Markov chains do not
include the ‘‘unobserved’’ FF memory). Flowering and
vegetative GUs, therefore, alternated along the sequences.
This alternation is superimposed upon the trend corre-
sponding to GU decrease in vigour along the sequences
(long ? medium ? short). This trend is highlighted by
the high transition probabilities pVFV where V is a vegeta-
tive state chosen from among L, M and S. Since for both
cultivars the transition probabilities between distinct veg-
etative states taken in order of decreasing vigour (mainly
Fig. 4 Fit of the empirical number of nodes distribution for extension
GUs by the mixture of long and medium state observation distribu-
tions: a Braeburn cultivar; b Fuji cultivar
Table 2 Means and standard deviations (indicated between brackets)
of the estimated observation distributions for the number of nodes of
extension GUs for the Braeburn and Fuji cultivars
Cultivar State
Long Medium
Braeburn 15.44 (8.85) 9.65 (3.46)
Fuji 17.57 (8.44) 8.09 (2.9)
Nodes corresponding to bud scars and whose axillary bud was not
visible were not counted
872 Trees (2012) 26:865–879
123
pLM and pMS) were nil or very low, the decrease in vigour of
the vegetative GUs was not direct but preferentially
required an intermediate flowering stage. This is also
clearly illustrated in the graph of possible transitions; see
the comment above and Fig. 2b for Braeburn and Fig. 3b
for Fuji).
Fig. 5 Fit of recurrence time
distributions for different
growth unit (GU) types for the
Braeburn cultivar. a and
b Distributions for long GUs
computed from estimated
variable- and first-order Markov
chains, respectively; c and
d distributions for medium GUs
computed from estimated
variable- and first-order Markov
chains, respectively; e and
f distributions for short and
flowering GUs, respectively
Trees (2012) 26:865–879 873
123
Fig. 6 Fit of recurrence time
distributions for different
growth unit (GU) types for the
Fuji cultivar. a and
b Distributions for long GUs
computed from estimated
variable- and first-order Markov
chains, respectively; c and
d distributions for medium GUs
computed from estimated
variable- and first-order Markov
chains, respectively; e and
f distributions for short and
flowering GUs, respectively
874 Trees (2012) 26:865–879
123
Analysing model outputs: between- and within-year
transitions and frequent GU successions
As a consequence of spring flowering followed by sym-
podial vegetative branching, transitions from a flowering
GU towards a vegetative GU (chosen from among long,
medium and short) corresponded almost exclusively to
within-year transitions while transitions from a vegetative
GU towards a flowering GU corresponded exclusively to
between-year transitions (Table 3). Transitions from a
vegetative GU towards another vegetative GU (often of the
same type) corresponded most often to between-year
transitions except in the case of two successive long GUs.
As a consequence of the integrative statistical modelling,
vegetative polycyclism corresponding to within-year tran-
sitions is a distinctive property of long GUs (Table 3). The
phenomenon occurred fairly rarely for Fuji (10% of long
GUs, i.e. 45 out of 444 GUs, and negligible for medium
and short GUs), while it was more frequent for Braeburn
(23% for long GUs, i.e. 110 out of 484, and negligible for
medium and short GUs).
As shown above on the basis of the memory trees (see
Fig. 2a for Braeburn and Fig. 3a for Fuji) and the restored
state sequences, one of the main differences between the
Braeburn and Fuji cultivars concerned short GUs. In par-
ticular, the transitions from a short GU towards another
short GU (mostly corresponding to between-year transi-
tions) were far more frequent for Fuji than Braeburn
(Table 3). As a consequence, the FSSF pattern, which
corresponds to biennial bearing, was far more frequent for
Fuji than for Braeburn (Table 4). Conversely, the FSF and
FSFSF patterns, which correspond to regular spring
flowering, were far more frequent for Braeburn than for
Fuji. These results, together with the low frequency of the
within-year SS pattern for Fuji cultivar (22 within-year SS
in Table 3, row ‘‘total short’’, column ‘‘short’’, compared
with 69 FSSF patterns in Table 4) illustrate the biennial
bearing behaviour of the Fuji cultivar. The maintenance of
Table 3 Between- and within-year transition counts extracted from the state sequences restored using the hidden variable-order Markov chains
estimated for Braeburn and Fuji cultivars
Between- and within-year transition counts
Long Medium Short Flowering Total
Braeburn memory
Long 78 100 2 4 17 4 277 2 374 110
Medium 0 0 7 5 0 0 559 0 566 5
Short 10 4 39 4 37 6 1,107 3 1,193 17
Initial flowering 0 93 0 108 6 582 0 0 6 783
Long flowering 0 116 1 136 0 0 0 0 1 252
Medium flowering 0 0 0 106 1 239 0 0 1 345
Short flowering 0 20 1 88 6 561 0 0 7 669
Fuji memory
Long 114 42 0 1 3 2 282 0 399 45
Medium 11 9 21 8 18 6 404 0 454 23
Initial short 30 0 18 1 277 1 648 0 973 2
Long short 1 0 0 0 0 0 1 0 2 0
Medium short 0 0 0 0 2 0 11 2 13 2
Short short 1 0 2 2 39 9 157 0 199 11
Flowering short 12 1 26 1 155 12 159 0 352 14
Total short 44 1 46 4 473 22 976 2 1,539 29
Initial flowering 1 23 5 101 19 527 0 0 25 651
Long flowering 1 100 0 68 0 6 0 0 1 174
Medium flowering 0 0 0 99 2 53 0 0 2 152
Short flowering 1 13 5 77 21 427 0 0 27 517
For each GU type, the first column corresponds to between-year transitions and the second column to within-year transitions. Transitions towards
the death state are not considered. The initial transient memories, which correspond to the first states in the sequences, are indicated in italics. The
row labelled ‘‘Total short’’ cumulates the counts for the initial transient ‘‘initial short’’ memory and the permanent second-order ‘‘long short’’,
‘‘medium short’’, ‘‘short short’’ and ‘‘flowering short’’ memories. This row is inserted for the Fuji cultivar to help the comparison with the first-
order ‘‘short’’ memory of the Braeburn cultivar
Trees (2012) 26:865–879 875
123
regular bearing over three successive years can be illus-
trated by patterns of length 5 starting and ending in the F
state (e.g. FSFSF); see Table 4. All these regular patterns
were more frequent for Braeburn than for Fuji.
Discussion
In this study we proposed an integrative statistical model
that provides a global and quantitative picture of the two-
scale structuring observed during apple tree ontogeny: a
coarse scale corresponding to the succession of two
developmental phases and a fine scale corresponding to the
alternation between flowering and vegetative GUs (Fig. 7).
The first phase, which is almost transient (a phase is said to
be transient if when leaving it, it is impossible to return to
it), was called the ‘‘adolescent’’ phase; see below for dis-
cussion of the chosen terminology. This corresponds to the
alternation between long and flowering GUs. The second
phase, hereafter referred to as the ‘‘adult’’ phase, includes
patterns of alternation between medium and flowering GUs
and between short and flowering GUs. This structuring is a
consequence of both the inclusion of second-order mem-
ories that specialize flowering GUs as a function of the
preceding vegetative GU, and the one-step integrative
statistical modelling. In contrast, with a two-step modelling
where long and medium GUs were defined on the basis of a
threshold on the number of nodes before modelling the GU
succession, the first transient phase did not emerge; the
results were similar with a model based on the GU length
instead of the number of nodes (results not shown). The
chosen modelling approach also led to an unexpected
characterization of the extension GUs in the adolescent and
adult phases since differentiation between the long and
medium GUs combined the number of nodes with struc-
tural properties:
• GUs with\15 nodes could be either labelled as long or
medium (but the proportion of long GUs increases with
the number of nodes; see the corresponding mixture of
observation distributions in Fig. 4) while GUs with
more than 15 nodes were systematically labelled as
long.
• Vegetative polycyclism was frequent for long GUs
whereas it was rare for medium GUs,
Table 4 Counts for frequent patterns* starting and ending in F extracted from the state sequences restored using the hidden variable-order
Markov chains estimated for the Braeburn and Fuji cultivars
3 GUs Braeburn Fuji 4 GUs Braeburn Fuji 5 GUs Braeburn Fuji
FSF 935 159 FSSF 5 69 FSFSF 281 16
FMF 397 155 FLLF 48 30 FMFMF 65 18
FLF 122 73 FSMF 19 13 FLFMF 61 14
FMLF 0 13 FSFMF 60 10
FMFSF 40 3
FLFLF 29 13
Total 1 1,454 387 90 147 553 93
Total 2 3,188 2,173 2,127 1,064 1,330 503
The cumulated counts for patterns of a given length starting and ending in F (Total 1) and for patterns of similar length (Total 2) are given. L, M,
S, F stands for long, medium, short and flowering, respectively
* (i.e. more than ten occurrences for at least one of the two cultivars)
Fig. 7 Schematic representation of apple tree ontogenetic gradients
for Braeburn (left part) and Fuji (right part). For both cultivars, the
gradual decrease in shoot growth with tree ageing results from a
reduction in neoformation which involves growth cessation during
annual growth cycle and generates polycyclic shoots during the
adolescent phase (in grey). Later, the reduction leads to transitions
towards medium and short GUs. The adult phase (in white)
corresponds to repetitions of short and medium GUs separated by
flowering GUs with specific repeated patterns depending on the
regular or alternate behaviour of the cultivar
876 Trees (2012) 26:865–879
123
• Transitions from a long GU (belonging to the adoles-
cent phase) to a short GU (belonging to the adult phase)
with an intermediate flowering GU were rare while
transition from a medium GU to a short GU (both
belonging to the adult phase) with an intermediate
flowering GU were frequent.
Both studied cultivars exhibited a similar macroscopic
structure with two successive phases. This demonstrates the
similarity of architectural organisation at the species level,
even though within-species differences also existed at the
growth unit scale (regular spring flowering for Braeburn vs.
biennial bearing for Fuji). The almost transient character of
the first phase is an expected biological result since it
includes the vegetative period that precedes first flowering
occurrences. However, first flowering represented by the LF
memory was also included in this phase. The increase in
annual shoot length observed during the first years of
development of forest trees (Guedon et al. 2007) and other
morphological changes that often occur during the first
phases of tree ontogeny (Nozeran 1984) were not observed in
our apple tree dataset. This results from the grafting of
mature material on rootstocks that is commonly employed in
fruit tree cultivation to promote early flowering (Hackett
1985). As a consequence, we considered that no sensu stricto
‘‘juvenile phase’’ was observed in the present study and that
the first identified phase should rather be qualified as ‘‘ado-
lescent’’, with reference to the terminology introduced by
Day et al. (1997). Moreover, axis bending was performed in
orchards to accelerate the occurrence of flowering (Lauri
2002). This agronomic manipulation may have enhanced the
occurrence of the first flowering and reduced the adolescent
phase. We expect that different conditions will mainly affect
the time spent in each phase as given by the transition dis-
tributions attached to the memories LF and MF (i.e. different
balances between pLFL and pLFM, and between pMFM and pMFS)
but not the main structure of the models.
One of the main outputs of the proposed modelling
approach is a characterisation of the two successive
developmental phases in terms of GU types and suc-
cession. The adolescent phase is characterized by the
occurrence of long GUs and includes GUs belonging to
polycyclic vegetative shoots. In the estimated model,
polycyclism distinguishes long from medium GUs. It can
therefore be used as a new criterion for classifying GUs
and could contribute to overcome the ambiguity between
GU categories on the basis of their sole number of
nodes. Polycyclism also appears to be an intermediate
developmental stage in the transition towards the adult
phase. This particular stage relies on the capability of the
shoot apical meristem, specific to perennial species, to
stop temporarily its organogenetic activity without
entering into floral differentiation. From an evolutionary
point of view, a propensity for polycyclism has been
interpreted as a strategy to maximize light interception
(Verdu and Climent 2007), while, from a physiological
point of view, it has been interpreted as resulting from
nutritional competition between organs (Barnola et al.
1990). Our results show that the propensity of a species
to develop polycyclic shoots can vary intra-specifically
since it differed here between the two studied cultivars.
While the repetitive nature of plant growth has been
described for a long time at the metamer scale (White
1979), the present study highlights the existence of repe-
ated patterns at more macroscopic scales that result from
the repetitive occurrence of flowering GUs in a perennial
plant. In the proposed model, flowering occurrences mark
transitions not only between the adolescent and adult
phases, but also between stages within the adult phase (i.e.
between stages characterized by medium and short vege-
tative GUs). Transitions due to flowering have in the past
been considered as abrupt, especially when ending the
juvenile phase (Hackett 1985). Our results show that, in a
perennial plant, abrupt changes following flowering occur
throughout tree life. Flowering thus appears to be a marker
of the transition between successive developmental stages
that are represented by discrete states in the proposed
model. The modelling here is consistent with the high
structuring property of GU types compared with the small
differences between years in the number of nodes they bear
(except for very long GUs that develop in the first 2 years
of growth). It is noteworthy that decrease in growth with
tree ageing is particularly rapid in the apple tree mainly
because of manipulation effects (rootstock and bending).
This contrasts with forest trees where long periods of sta-
bility have been observed over decades separated by abrupt
changes (Guedon et al. 2007). In addition, a minimal and
recurrent stage, represented by the SF memory, was
reached, corresponding to a ‘‘minimal architectural unit’’
(Barthelemy and Caraglio 2007). In our dataset, flowering
occurrence after a minimum number of nodes can also be
interpreted with respect to alternate flowering which is a
specificity of perennial plants resulting from the absence of
flowering induction in particular years. The particular sta-
tus of flowering in tree ontogeny may arise from the high
carbon cost of flower and fruit production (Bustan and
Goldschmidt 1998) and competition with current vegeta-
tive growth (Cannell 1985; Berman and DeJong 1997). The
terminal position of flowering and subsequent sympodial
branching may also be involved in the reduction in vege-
tative growth since they induce a rupture in vascular con-
nections and probably in hydraulic conductance, and this is
assumed to be responsible for age-related decline in trees
(Martinez-Vilalta et al. 2007).
From a methodological point of view, the present
study paves the way to further investigations. First-order
Trees (2012) 26:865–879 877
123
Markov chains were applied in two other studies to
analyse successions of plant entities based on data col-
lected during follow-up that lasted several years (Mail-
lette 1990; Sterck et al. 2003). For instance, in
Maillette’s work on mountain white birches, the states
are long-shoot production, short-shoot production, sum-
mer dormancy (ecodormancy) and death. The states
identified in our context can be viewed as a retrospective
analogue of the meristem states defined by Maillette in a
prospective framework except for summer dormancy
which was not included in our modelling. In our
approach, some states (long and medium) were deduced
from an optimization based both on measurements
(number of nodes) and the properties of the succession
of GUs along the sequences. The focus was also dif-
ferent since our main objective was to identify the rules
governing the succession of differentiation states along
GU sequences while in Maillette (1990) and Sterck et al.
(2003), the aim was rather to characterise the demogra-
phy of plant entities during development. Hence, an
interesting avenue for future investigations would be to
combine the two approaches and study the demography
of plant entities based on sophisticated stochastic models.
In the present analysis, the trees were explored
exclusively through terminal transitions along either
monopodial or sympodial shoots. In contrast, transitions
resulting from lateral branching were modelled only
globally by the estimated initial probabilities of the
hidden variable-order Markov chains. But, the branching
structures resulting from lateral transitions along long
and medium GUs have been widely explored in previous
studies using mainly hidden semi-Markov chains; see
Guedon et al. (2001), Renton et al. (2006) and references
therein. A simulation system based on a multiscale sto-
chastic model combining a first-order Markov chain for
modelling GU succession and hidden semi-Markov
chains for modelling GU branching structure was pro-
posed by Costes et al. (2008); see also Lopez et al.
(2008) for a similar approach in the peach tree case. A
direct output of the present study would be to replace the
first-order Markov chain by a variable-order Markov
chain to better model growth phases and repeated pat-
terns in GU succession. Such simulation systems com-
bining sophisticated stochastic models for modelling tree
topology with different types of mechanistic models (e.g.
mechanical model for branch bending or model for
carbohydrate partitioning) open new perspectives for in
silico investigations of agronomical scenarios. Similar
approaches could be applied to other perennial species,
including monopodial and sympodial species.
Acknowledgments This research was funded by both the INRA
Genetic and Breeding Department and the CIRAD Bios Department.
We thank Michael Renton for his contribution in the first steps of GU
analyses and Pierre-Eric Lauri for helpful comments.
Appendix A: Selection of the memories of a variable-
order Markov chain
The order of a Markov chain can be estimated using the
Bayesian information criterion (BIC). For each possible
order r, the following quantity is computed
BIC rð Þ ¼ 2 log Lr � Jr J � 1ð Þ log n; ð1Þ
where Lr is the likelihood of the rth-order estimated Mar-
kov chain for the observed sequences, Jr(J - 1) is the
number of independent transition probabilities of a J-state
rth-order Markov chain and n is the cumulated length of
the observed sequences. The principle of this penalized
likelihood criterion consists in making a trade-off between
an adequate fitting of the model to the data [given by the
first term in (1)] and a reasonable number of parameters to
be estimated (control by the second term, the penalty term).
In practice, it is infeasible to compute a BIC value for each
possible variable-order Markov chain of maximum order
r B R since the number of hypothetical memory trees is
very large. An initial maximal memory tree is thus built
combining criteria relative to the maximum order and to
the minimum count of memory occurrences in the observed
sequences. This memory tree is then pruned, using a two-
pass algorithm which is an adaptation of the Context-tree
maximizing algorithm (Csiszar and Talata 2006): a first
dynamic programming pass, starting from the terminal
vertices and progressing towards the root vertex for com-
puting the maximum BIC value attached to each sub-tree
rooted in a given vertex, is followed by a second tracking
pass starting from the root vertex and progressing towards
the terminal vertices for building the memory tree.
Appendix B: Definition of parametric observation
distributions for hidden variable-order Markov chains
In the case of a count variable such as the number of nodes
of a GU, the observation distributions are parametric dis-
crete distributions chosen from among binomial distribu-
tions, Poisson distributions and negative binomial
distributions with an additional shift parameter d.
The binomial distribution with parameters d, n and
p (q = 1 - p), B(d, n, p) where 0 B p B 1, is defined by
bj yð Þ ¼ n� dy� d
� �py�dqn�y; y ¼ d; d þ 1; . . .; n:
The Poisson distribution with parameters d and k, P(d,
k), where k is a real number (k[ 0), is defined by
878 Trees (2012) 26:865–879
123
bj yð Þ ¼ e�kky�d
y� dð Þ! ; y ¼ d; d þ 1; . . .
The negative binomial distribution with parameters d, r
and p, NB(d, r, p), where r is a real number (r [ 0) and
0 \ p B 1, is defined by
bj yð Þ ¼ y� d þ r � 1
r � 1
� �prqy�d; y ¼ d; d þ 1; . . .
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