ORIGINAL PAPER Deciphering the ontogeny of a sympodial tree Evelyne Costes • Yann Gue ´don Received: 26 April 2011 / Revised: 18 August 2011 / Accepted: 23 November 2011 / Published online: 13 December 2011 Ó 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 Barthe ´le ´my 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. Gue ´don (&) 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
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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
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