Chapter 8 Decision Trees

Decision trees, also known as classification and regression tree (CART) models, are tree-based methods for supervised machine learning. Simple classification trees and regression trees are easy to use and interpret, but are not competitive with the best machine learning methods. However, they form the foundation for ensemble models such as bagged trees, random forests, and boosted trees, which although less interpretable, are very accurate.

CART models segment the predictor space into \(K\) non-overlapping terminal nodes (leaves). Each node is described by a set of rules which can be used to predict new responses. The predicted value \(\hat{y}\) for each node is the mode (classification) or mean (regression).

CART models define the nodes through a top-down greedy process called recursive binary splitting. The process is top-down because it begins at the top of the tree with all observations in a single region and successively splits the predictor space. It is greedy because at each splitting step, the best split is made at that particular step without consideration to subsequent splits.

The best split is the predictor variable and cutpoint that minimizes a cost function. The most common cost function for regression trees is the sum of squared residuals,

\[RSS = \sum_{k=1}^K\sum_{i \in A_k}{\left(y_i - \hat{y}_{A_k} \right)^2}.\]

For classification trees, it is the Gini index,

\[G = \sum_{c=1}^C{\hat{p}_{kc}(1 - \hat{p}_{kc})},\]

and the entropy (aka information statistic)

\[D = - \sum_{c=1}^C{\hat{p}_{kc} \log \hat{p}_{kc}}\]

where \(\hat{p}_{kc}\) is the proportion of training observations in node \(k\) that are class \(c\). A completely pure node in a binary tree would have \(\hat{p} \in \{ 0, 1 \}\) and \(G = D = 0\). A completely impure node in a binary tree would have \(\hat{p} = 0.5\) and \(G = 0.5^2 \cdot 2 = 0.25\) and \(D = -(0.5 \log(0.5)) \cdot 2 = 0.69\).

CART repeats the splitting process for each child node until a stopping criterion is satisfied, usually when no node size surpasses a predefined maximum, or continued splitting does not improve the model significantly. CART may also impose a minimum number of observations in each node.

The resulting tree likely over-fits the training data and therefore does not generalize well to test data, so CART prunes the tree, minimizing the cross-validated prediction error. Rather than cross-validating every possible subtree to find the one with minimum error, CART uses cost-complexity pruning. Cost-complexity is the tradeoff between error (cost) and tree size (complexity) where the tradeoff is quantified with cost-complexity parameter \(c_p\). The cost complexity of the tree, \(R_{c_p}(T)\), is the sum of its risk (error) plus a “cost complexity” factor \(c_p\) multiple of the tree size \(|T|\).

\[R_{c_p}(T) = R(T) + c_p|T|\]

\(c_p\) can take on any value from \([0..\infty]\), but it turns out there is an optimal tree for ranges of \(c_p\) values, so there are only a finite set of interesting values for \(c_p\) (James et al. 2013) (Therneau and Atkinson 2019) (Kuhn and Johnson 2016). A parametric algorithm identifies the interesting \(c_p\) values and their associated pruned trees, \(T_{c_p}\). CART uses cross-validation to determine which \(c_p\) is optimal.


James, Gareth, Daniela Witten, Trevor Hastie, and Robert Tibshirani. 2013. An Introduction to Statistical Learning: With Applications in R. 1st ed. New York, NY: Springer.

Kuhn, Max, and Kjell Johnson. 2016. Applied Predictive Modeling. 1st ed. New York, NY: Springer.

Therneau, Terry, and Elizabeth Atkinson. 2019. An Introduction to Recursive Partitioning Using the Rpart Routines. Boca Raton, Florida: Chapman; Hall/CRC.