  We provide finite sample guarantees for the classical Chow-Liu algorithm
(IEEE Trans.~Inform.~Theory, 1968) to learn a tree-structured graphical model
of a distribution. For a distribution $P$ on $\Sigma^n$ and a tree $T$ on $n$
nodes, we say $T$ is an $\varepsilon$-approximate tree for $P$ if there is a
$T$-structured distribution $Q$ such that $D(P\;||\;Q)$ is at most
$\varepsilon$ more than the best possible tree-structured distribution for $P$.
We show that if $P$ itself is tree-structured, then the Chow-Liu algorithm with
the plug-in estimator for mutual information with $\widetilde{O}(|\Sigma|^3
n\varepsilon^{-1})$ i.i.d.~samples outputs an $\varepsilon$-approximate tree
for $P$ with constant probability. In contrast, for a general $P$ (which may
not be tree-structured), $\Omega(n^2\varepsilon^{-2})$ samples are necessary to
find an $\varepsilon$-approximate tree. Our upper bound is based on a new
conditional independence tester that addresses an open problem posed by
Canonne, Diakonikolas, Kane, and Stewart~(STOC, 2018): we prove that for three
random variables $X,Y,Z$ each over $\Sigma$, testing if $I(X; Y \mid Z)$ is $0$
or $\geq \varepsilon$ is possible with $\widetilde{O}(|\Sigma|^3/\varepsilon)$
samples. Finally, we show that for a specific tree $T$, with $\widetilde{O}
(|\Sigma|^2n\varepsilon^{-1})$ samples from a distribution $P$ over $\Sigma^n$,
one can efficiently learn the closest $T$-structured distribution in KL
divergence by applying the add-1 estimator at each node.
