Splitting criterion of classification tree: Does the growth process come naturally to a stop? With respect to growing a classification tree: Does growing with Gini or Cross-entropy (CE) imply we would grow the tree until every leaf is pure (in case of no other stopping criteria)? Put differently: is Gini/CE always benefitted (reduced) by an additional split if the nodes are not completely pure yet?
And in contrast, is there for the misclassification error such a situation, where nodes are not pure yet but we don't continue growing as a further split would not reduce the misclassification error?

Suppose your model has 1 feature. You've been constructing a tree and you've reached a node where you have 6 samples (3 positives and 3 negatives) in your training data. In this scenario, for these particular samples, the feature values are identical. Any split you choose on your feature will put the samples in the same child node, and the purity of that child node will be the same as the parent node.
