  Given side information that an Ising tree-structured graphical model is
homogeneous and has no external field, we derive the exact asymptotics of
learning its structure from independently drawn samples. Our results, which
leverage the use of probabilistic tools from the theory of strong large
deviations, refine the large deviation (error exponents) results of Tan,
Anandkumar, Tong, and Willsky [IEEE Trans. on Inform. Th., 57(3):1714--1735,
2011] and strictly improve those of Bresler and Karzand [Ann. Statist., 2020].
In addition, we extend our results to the scenario in which the samples are
observed in random noise. In this case, we show that they strictly improve on
the recent results of Nikolakakis, Kalogerias, and Sarwate [Proc. AISTATS,
1771--1782, 2019]. Our theoretical results demonstrate keen agreement with
experimental results for sample sizes as small as that in the hundreds.
