  Knowing the causal structure of a system is of fundamental interest in many
areas of science and can aid the design of prediction algorithms that work well
under manipulations to the system. The causal structure becomes identifiable
from the observational distribution under certain restrictions. To learn the
structure from data, score-based methods evaluate different graphs according to
the quality of their fits. However, for large, continuous, and nonlinear
models, these rely on heuristic optimization approaches with no general
guarantees of recovering the true causal structure. In this paper, we consider
structure learning of directed trees. We propose a fast and scalable method
based on Chu-Liu-Edmonds' algorithm we call causal additive trees (CAT). For
the case of Gaussian errors, we prove consistency in an asymptotic regime with
a vanishing identifiability gap. We also introduce two methods for testing
substructure hypotheses with asymptotic family-wise error rate control that is
valid post-selection and in unidentified settings. Furthermore, we study the
identifiability gap, which quantifies how much better the true causal model
fits the observational distribution, and prove that it is lower bounded by
local properties of the causal model. Simulation studies demonstrate the
favorable performance of CAT compared to competing structure learning methods.
