GreenLab Course
Development
Factorisation
Substructure construction
Buds, Phytomers and Substructures

From the botanical description of the plant, we know that phytomers and buds are the elementary bricks of
plant structure.
They are derived from buds and build the axes.
Let P be the maximum number of physiological ages in the plant.
In practice, P is generally small (P < 5).
At growth cycle t, a metamer (i.e. a microstate) is characterized by its physiological age p, and the physiological age of its axillary branches q, with q <= p, and its chronological age n.
It is denoted by m^{t}_{p,q}(n).
These three indices p, q, and n are sufficient to describe all the phytomers and their number grows linearly with t.
A bud is only characterized by its physiological age p and is denoted by s_{p}.
The terminal bud of a plant axis produces different kinds of metamers bearing axillary buds of various physiological ages. These buds themselves give birth to axillary branches and so on.
A substructure is the complete plant structure that is generated after one or more cycles by a bud.
In the deterministic case, all the subestructures with the same physiological and chronological ages are identical if they have developed at the same moment in the tree architecture.
At cycle t, a substructure is thus characterized by its physiological age p and its chronological age n.
It is denoted by S^{t}_{p}(n).
Since the physiological age of the main trunk is 1, at growth cycle t, the substructure of physiological age 1 and of chronological age t, S^{t}_{1}(t),represents the whole plant.
Subestructures encoding (Drawings C. Loi, P.H. CournĂ¨de, ECOLE CENTRALE PARIS)
 Factorizing plant development and its subestructure inductive construction: example
for a plant off three physiological ages.
Blue, green and red are respectively the colours corresponding to physiological age 1, 2 and 3.
Circles materialize the buds, while rectangles the phytomers.
S_{1}(2), while S_{2}(2) and S_{3}(2) are substructures of age 2 at physiological ages 2 and 3, respectively. They may both appear in the tree at chronological age 3.
The total number of different substructures in a plant of chronological age t is very small, usually fewer than 30, even if the total number of organs is high.
Substructures and phytomers are repeated many times in the tree architecture, but they need to be computed only once for each kind.
The substructure construction equation

The concatenation operator to describe the organization of plant phytomers and substructures
and deduce their construction at growth cycle t by induction is as follows:
 Substructures of chronological age zero are buds:
S^{t}_{p}(0) = s_{p}  If all substructures of chronological age n  1 are built, we deduce the substructures
of chronological age n as:
S^{t}_{p}(n) = [ ∏_{p≤q≤P} ( m^{t}_{p,q}(n) )^{up,q(t+1n)} ( S^{t}_{q}(n1) )^{bp,q(t+1n)} ] S^{t}_{p}(n1) (equation 6)
for all (p,q) such as p ≤ P and (p ≤ q ≤ P)
where

u_{p,q} corresponds to the number of metamers m_{p,q}(t) in growth units of
physiological age p appearing at growth cycle t
b_{p,q} corresponds to the number of axillary substructures of physiological age q in growth units of physiological age p that appeared at growth cycle t
These sequences can be deterministic or stochastic.
This construction equation is used to count the metamers of each substructure.
Formula interpretation
∏_{p≤q≤P} ( m^{t}_{p,q}(n) )^{up,q(t+1n)}
stands for the existing old phytomers on the substructure main axis (the base growth unit) ∏_{p≤q≤P} ( S^{t}_{q}(n1) )^{bp,q(t+1n)} stands for the lateral substructures borne by the base growth unit (they are one cycle younger) S^{t}_{p}(n1) stands for the substructure grown from the apical bud of the base growth unit (also one cycle younger). 
(Drawing M. Jaeger, CIRAD) 
Note
∏ is the product operator, used in mathematics to represent the product of a bunch of terms. ∏_{k=2,5}(k) = 2 x 3 x 4 x 5 = 120
Bibliography
Yan H.P., Barczi J.F. , de Reffye P., and Hu B.G. 2002. Fast Algorithms of Plant Computation Based on Substructure Instances. International Conferences in Central Europe on Computer Graphics, Visualization and Computer Vision 3/10 (2002) pp. 145153 (access to paper and pdf)