Extra-trees random split

Pick the threshold at random

A standard decision tree searches every candidate threshold for the best split. Extremely randomized trees skip that search: for a chosen feature they draw one threshold uniformly between the feature's minimum and maximum in the node, and keep the better of the random choices across features.

Implement extra_tree_split(x, y, seed) for one feature x and binary labels y in {0, 1}. Return (threshold, gain).

1. Let lo = x.min() and hi = x.max(). If lo == hi, return (None, 0.0).
2. Create rng = np.random.default_rng(seed) and set threshold = rng.uniform(lo, hi).
3. Send rows with x <= threshold left and the rest right.
4. Gain is the Gini reduction: gini(y) − (n_L / n) · gini(y_L) − (n_R / n) · gini(y_R), where gini(p) = 1 − Σ p_c². If one side is empty, the gain is 0.0.

Constraints

• x and y must have the same length, otherwise raise ValueError.
• Do not modify the inputs.

Hints