I've have this question and I'm completely lost. Would really appreciate an answer to this question. Thank you.
Let $D_a$ denote the training data set and let $T$ be a decision tree trained on $D_a$ through an error measuring function $E$ (e.g. $SS_{Res}$ on a regression tree or crossentropy on a classification tree). Let $T'$ denote the resulting tree after a split on $T$, and suppose that $E(D_a, T) = E(D_a, T')$. Should the splitting process be stopped? Justify your answer.
You may have a result that states that for most "error-measuring functions" including the examples you gave, we have $E(D_a,T) \ge E(D_a,T')$ for any split $T'$ of $T$. That is, splitting never makes things worse (on the given data). In your situation, splitting did not improve anything, so we tend to stay with the simpler tree (before the split) if there are no better splits elsewhere.