  We develop optimal algorithms for learning undirected Gaussian trees and
directed Gaussian polytrees from data. We consider both problems of
distribution learning (i.e. in KL distance) and structure learning (i.e. exact
recovery). The first approach is based on the Chow-Liu algorithm, and learns an
optimal tree-structured distribution efficiently. The second approach is a
modification of the PC algorithm for polytrees that uses partial correlation as
a conditional independence tester for constraint-based structure learning. We
derive explicit finite-sample guarantees for both approaches, and show that
both approaches are optimal by deriving matching lower bounds. Additionally, we
conduct numerical experiments to compare the performance of various algorithms,
providing further insights and empirical evidence.
