Example 1
The first example represents shoot growth as the repeated production of new structural modules or metamers (each shown as a coloured blob).

In a plant during the vegetative (that is, non-reproductive) phase of its lifecycle each metamer could be thought of as a section of stem to which a leaf and an axillary bud is attached.

Every so often a branch develops as one of these buds begins to grow out as a new shoot.

The animation shows the branching pattern developing and progressive senescence (yellow blobs) following the wave of growth and branching.

Example 2
The second example follows the pattern of the first, but now the senescent modules are lost, by shedding, death and decay.

This is what happens in creeping plants like clover.

As the wave of senescence and death moves through the plant body, individual branches become isolated and continue to develop as if they were separate individuals.

These individuals (called ramets) are genetically identical and constitute a clonal population.

Some clonal plants achieve immense ages.

Notice also that this kind of horizontal perenniality allows members of the clone to move around the environment - an effective way of finding and systematically exploiting new resources.

Example 3
This is similar in principle to horizontal perenniality, but here senescent modules, instead of disappearing, become lignified and persist as woody tissue (brown blobs).

The result is a tree - an example of vertical perenniality.

Example 4
Finally, example 4 shows what might happen if the kind of structural module changes during growth.

In this case a terminal flower (blue) is produced at the apex of each branch.

Again a wave of senescence advances through the plant body, but this time it completely overtakes the production of new modules.

The result is monocarpy - suicidal reproduction.

More on this in:
H Thomas (1994) Aging in the plant and animal kingdoms - the role of cell death. Reviews in Clinical Gerontology 4: 5-20
H Thomas, H Ougham, H M Thomas (2000) Annuality, perenniality and cell death. Journal of Experimental Botany 51: 1-8
H Thomas (2003) Do green plants age and if so how? Topics in Current Genetics 3: 145-171
S Munné-Bosch (2007) Aging in perennials. Critical Reviews in Plant Sciences, 26:123–138

Modelling plant development
By thinking of development and lifespan in terms of patterns of growth and senescence, it's quite straightforward to build useful mathematical models.

Such a model may include equations for the rate of production of new modules and of senescence.

It will have rules for when branching occurs and when senescence is initiated.

These will be based on established physiological principles; for example, an axillary bud will become activated when sufficiently remote from the root, or from the inhibitory influence of the terminal bud on the main axis.

There are well established mechanisms, based on hormonal gradients, that can put this on a quantitative basis.

Similarly, senescence and recycling are triggered when the number and sink strength of modules exceed the capacity of nutrient uptake to meet the demand.

A typical plant development model will bring all these elements together in a single mathematical representation that can have valuable descriptive and predictive power.

Further reading:
FL Milthorpe, J Moorby (1974) An Introduction to Crop Physiology. CUP
J Thornley, J France (2007) Mathematical Models in Agriculture. OUP
P Prusinkiewicz, A Lindenmayer (1990) The Algorithmic Beauty of Plants. Springer-Verlag (see also the excellent Algorithmic Botany website of the University of Calgary)