  Label tree-based algorithms are widely used to tackle multi-class and
multi-label problems with a large number of labels. We focus on a particular
subclass of these algorithms that use probabilistic classifiers in the tree
nodes. Examples of such algorithms are hierarchical softmax (HSM), designed for
multi-class classification, and probabilistic label trees (PLTs) that
generalize HSM to multi-label problems. If the tree structure is given,
learning of PLT can be solved with provable regret guaranties [Wydmuch et.al.
2018]. However, to find a tree structure that results in a PLT with a low
training and prediction computational costs as well as low statistical error
seems to be a very challenging problem, not well-understood yet.
  In this paper, we address the problem of finding a tree structure that has
low computational cost. First, we show that finding a tree with optimal
training cost is NP-complete, nevertheless there are some tractable special
cases with either perfect approximation or exact solution that can be obtained
in linear time in terms of the number of labels $m$. For the general case, we
obtain $O(\log m)$ approximation in linear time too. Moreover, we prove an
upper bound on the expected prediction cost expressed in terms of the expected
training cost. We also show that under additional assumptions the prediction
cost of a PLT is $O(\log m)$.
