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Pruning is one of the key procedures in training decision tree classifiers. It removes trivial rules from the raw knowledge base built from training examples, in order to avoid over-using noisy, conflicting, or fuzzy inputs, so that the refined model can generalize better with unseen cases. In this paper, we present a number of properties of k-norm pruning, a recently proposed pruning algorithm, which has clear theoretical interpretation. In an earlier paper it was shown that k-norm pruning compares very favorably in terms of accuracy and size with minimal cost-complexity pruning and error based pruning, two of the most cited decision tree pruning methods; it was also shown that k-norm pruning is more efficient, at times orders of magnitude more efficient than minimal cost-complexity pruning and error based pruning. In this paper, we demonstrate the validity of the k-norm properties through a series of theorems, and explain their practical significance.