# Branching out with math

## Branching out with math McGill University

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February 19, 2004 - Volume 36 Number 11
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# Branching out with math

In the winter, the complex arrangement of a tree's branching pattern is plainly visible. During the growing season, however, these branches push forth the tree's leaves and thus influence the plant's light interception. By examining the mathematics behind a plant's branching pattern, professor Pierre Dutilleul, a statistician and a member of the Department of Plant Science, seeks to improve our understanding of a plant's light interception.

Plant science professor and statistician Pierre Dutilleul
Owen Egan

As an applied statistician, Dutilleul's study of plant branching patterns is a natural extension of his research. "I always sought to develop links between statistics and the life sciences," said Dutilleul. "This bridge-building is a main motivation in my work." Many things in nature can be modelled using mathematics and statistics. Light interception by plant canopies is no exception.

"A more accurate and complete model will improve our understanding of photosynthesis on a whole-plant basis and, eventually, our ability to select plants that perform better in terms of light interception," said Dutilleul. "This is important for questions of food production and environmental protection." However, some key facts about a plant's branching pattern were missing from the existing models.

A plant's light interception depends on its branching pattern. For example, imagine two trees with the same leaf area. One has its leaves arranged one on top of the other in a narrow column; the other has its leaves spread out, filling up much more space. The light intercepted by the two trees differs because of their different branching patterns.

To understand and quantify the complexity of branching patterns, fractals — detailed repeated geometric structures — are essential. For example, examining a snowflake, one notices that there are many tiny crystals on each of its arms. Zooming in further onto one of these arms, there are even more, tinier arms. The shape will continually replicate itself on a smaller and smaller scale. Snowflakes are just one of the many examples of fractals found in nature. Although on a different scale, tree branching also exhibits similar properties.

When Dutilleul and his colleagues, PhD student Kayhan Foroutan-pour and Professor Donald Smith of the Department of Plant Science, began examining the structure of plant canopies, they used two-dimensional photography to gather data. From photographs of leafless soybean plants, the team estimated each plant's fractal dimension and the complexity of the branching pattern. Later, they incorporated the new variable into their mathematical model of the plant canopies' light interception, which made it more accurate.

The success of two-dimensional photography motivated Dutilleul to extend his research into three dimensions. In doing so, he coordinated the creation of the CT Scanning Laboratory for agricultural and environmental research at the Macdonald Campus. The lab, the first of its kind in Eastern Canada, allows for detailed, non-destructive study of plants, soils and animals.

A helical computed tomography (CT) scanner, essentially an X-ray tube that rotates 360 degrees around the specimen, indirectly measures the density of each point of the specimen by assigning it a CT number. Parts of the specimen with similar densities have similar CT numbers.

After scanning a plant, such as a young cedar, a computer converts its CT number data into a digital three-dimensional model. As leaves and branches yield different CT numbers, the leaves can be removed from the digital model. Special software, developed by Dutilleul's team, estimates the fractal dimension of the remaining branches. The final step remains to incorporate the improved fractal dimension estimate into the mathematical model explaining a plant canopy's light interception — making it more accurate and complete.

The next time you stroll through the park in the winter, take a moment to appreciate a tree's branching pattern. As it turns out, the way a plant branches determines how much light it will intercept, and mathematics holds the key to understanding this fact of nature.

McGill's SPARK program (Students Promoting Awareness of Research Knowledge) is funded by NSERC and run by the Faculty of Education, VP Research Office and the University Relations Office. See Spark website.

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