How CART Decision Trees Model Interactions#

Stage 7 · 🌳 Decision Trees · Lesson 47 of 56 · advanced

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Important

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What is an interaction?#

An interaction occurs when the effect of one feature on the outcome depends on the value of another. Airport surcharges might make distance matter more for airport trips than for others; a drug might help one age group and harm another. In each case you cannot describe the effect of one variable without knowing the other — the two interact.

The regression chore#

Ordinary linear regression cannot see interactions on its own. Its form is strictly additive — each feature contributes its coefficient times its value, independently — so the effect of one feature is the same regardless of the others. To model an interaction you must manually add a product term (\(x_1 \times x_2\)), and you have to know in advance which interactions to include. Miss one, and the model is blind to it.

Trees get them free#

Decision trees capture interactions automatically, as a byproduct of their structure. Because splits are nested, a split on one feature can be followed by different splits on another in different branches — so the effect of the second feature genuinely differs depending on the first. A tree might split on trip type, then split on distance only in the airport branch: exactly an interaction between type and distance, discovered without anyone specifying it. Each split is an effect conditional on all the splits above it.

Depth and order#

This is why tree depth matters for expressiveness. A one-split (depth-1) tree captures only a single feature’s main effect; each additional level lets the tree condition on one more feature, so deep trees can represent high-order interactions among many variables. The first split naturally falls on the feature with the strongest overall (main) effect, with interactions emerging below it. This effortless interaction modelling — together with the piecewise structure of the last lesson — is what lets a single tree describe complex, realistic patterns, and it powers the cluster-explanation uses that close this stage.

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