The Many Dimensions of Tree Trunks

measuring tree height
Cory Tucker measures tree height. Photo by Dale Brockway, USFS.

The taller a tree grows, the wider its trunk becomes. “It’s a fundamental relationship,” says U.S. Forest Service research ecologist Dale Brockway.

Scientists use math to describe this height-diameter relationship. But one of the mathematical constants – a scaling exponent of 2/3 – is not always so constant, according to a recent study by Brockway and Xiongwen Chen. Chen led the study, which was published in the Journal of Plant Studies.

“In many scenarios, the 2/3 or 0.67 scaling exponent is valid,” says Brockway. Metabolic ecology researchers identified the scaling exponent. The exponent describes an optimum design for tree vascular systems.

“However, environmental conditions can influence the scaling exponents observed in the field,” says Brockway.

The researchers measured hundreds of trees across six locations in Alabama and Florida. They measured longleaf pine at the southern, central, and northern parts of the species’ natural range. The three locations receive different amounts of precipitation.

“Water availability affects the height-diameter relationship in longleaf pine,” says Brockway. “Water favors a higher exponent, 0.71 for example – which means tree height is greater, relative to the diameter.” In the drier forest, the exponent was 0.42. Lower exponents indicate a shorter height, relative to the diameter.

nyssa aquatica
Water tupelo forms buttresses (enlarged bases). Buttresses affect the height-diameter relationship. Photo by Miguel Vieira, CC 2.0.

The results were not a surprise to Brockway, who has been examining trees for decades. “Like all organisms, trees respond to their environment,” he says. “Our results are an example of species adaptation. Trees may have the capacity to adjust to spatial and temporal variation in local environments.”

The scientists also measured height-diameter relationships for water tupelo, bald cypress, red maple, and river birch. All four species are adapted to swampy conditions.

Red maple and river birch had the expected height-diameter relationships, close to 0.67.

However, scaling exponents for water tupelo and bald cypress were significantly less than 0.67. Water tupelo had a scaling exponent of 0.31, and bald cypress was 0.52.

Both species develop buttresses. Buttresses help trees acquire oxygen in flooded soils and may protect from toppling. Buttresses affect the height-diameter relationship by diverting internal carbohydrate resources away from height growth.

However, in many cases, the 2/3 scaling exponent describes tree growth very well.

Longleaf pine
Environmental conditions – especially water availability – affect the height-diameter relationship. Photo by Albert Herring, CC 2.0.

“There appears to be substantial variability,” says Brockway. “The scaling exponent is a metric that helps us describe how trees are responding to their environment. It may also help us understand how tree species are adapting to varied and changing conditions.”

Foresters use scaling metrics to estimate growth, productivity, and yield.

“It is important to appreciate the natural variation in scaling exponents for tree species,” says Brockway. “Assuming a single scaling exponent across all site types could result in less accurate estimates of tree growth. It is better to recognize that high and low levels of water availability, as well as buttress formation, can lead to scaling exponents that significantly depart from 2/3.”

Read the full text of the study.

For more information, email Dale Brockway at

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