# Understanding the $\alpha$-regularity assumption for trees

In this paper, definition 4 claims that a

tree grown by recursive partitioning is $$\alpha$$-regular for some $$\alpha>0$$ if each split leaves at least a fraction $$\alpha$$ of the available training examples on each side of the split, and moreover, the trees are fully grown to depth $$k$$ for some $$k \in \mathbb{N}$$ (i.e. there are between $$k$$ and $$2k-1$$ observations in each terminal node of the tree.

the first part of the definition makes sense to me, but I am not sure I understand how they conclude that there are between $$k$$ and $$2k-1$$ observations in a terminal node. Say you train a tree with $$n$$ observations, and you choose a depth $$k$$, then based on the $$\alpha$$-regularity, and assuming $$\alpha < 0.5$$ the minimum number of observations in a node at depth $$k$$ is going to be $$n\alpha^k$$, and the largest is $$n(1-\alpha)^k$$ - so I am not sure how they are able to conclude that the size of the node is between $$k$$ and $$2k-1$$.

update: my best guess is that they didn't mean a depth of $$k$$, but rather they require that the tree is grown fully, and since $$\alpha$$ is a proportion, it can at most be $$0.5$$, and so the number of observations in a terminal leaf will be between $$k$$ and $$2k-1$$ for some $$k$$. As the number of data points gets larger, this requires that the tree to become deeper since (I assume) $$k$$ is kept fixed