Can the size of a tree trunk help us estimate its timber volume?

• Measuring trunk girth is relatively straightforward.
• Volume is harder to estimate by looking at a tree.
• We will use regression to study the relationship.

We use R’s built-in trees dataset, which has measurements for 31 black cherry trees.

• Girth: trunk diameter in inches.
• Height: tree height in feet.
• Volume: timber volume in cubic feet.

Here, volume means the dataset’s measured timber volume, not every piece of wood in the whole tree.

Each point represents one tree. Is there a pattern?

We use trunk diameter to predict timber volume:

\[ Volume_i = \beta_0 + \beta_1 Girth_i + \varepsilon_i \]

• \(\beta_0\): intercept.
• \(\beta_1\): expected change in volume for one more inch of trunk diameter.
• \(\varepsilon_i\): differences the model does not explain.

model <- lm(Volume ~ Girth, data = trees)
round(coef(model), 2)

## (Intercept)       Girth 
##      -36.94        5.07

The fitted model gives a prediction using:

\[ \widehat{Volume} = \widehat{\beta}_0 + \widehat{\beta}_1 \times Girth \]

What volume does it predict for a tree with a 14-inch trunk diameter?

##     1 
## 33.98

The result is an estimate in cubic feet.

Rotate the plot and hover over a point to inspect a tree.

• Trees with wider trunks generally have more timber volume in this dataset.
• The fitted slope is about 5.07 cubic feet per additional inch of trunk diameter.
• For a 14-inch trunk, the model predicts about 33.98 cubic feet.

This is a relationship in the data. It does not prove that diameter alone determines volume.

• The dataset contains only 31 black cherry trees.
• Trees with the same diameter can have different heights and volumes.
• Our simple model uses diameter but leaves out height.
• We should be careful when predicting far outside the diameters in this dataset.

Data source: R’s built-in trees dataset. Charts made with ggplot2 and plotly.