A Proofs A . 1 Proof of Theorem 1 ( Sufficient number of samples ) Lemmas

Lemmas 1, 2, 3 together with Lemma 4 give the required number of samples for exact structure recovery when observations from a hidden model are given. To analyze the error event we use the "Two Trees Lemma” of Bresler and Karzand [75, Lemmas 10.1, 10.2]. For two different spanning trees on same set of nodes. Informally, if two maximum spanning trees T, T′ have a pair of nodes connected in a different way then there exist at least one edge in ET which does not exist in ET′ and vice versa. Lemma 1. Let f = (w, w̄) be an edge such that f ∈ T and f / ∈ TCL † . Then there exists an edge g ∈ TCL † and g / ∈ T such that f ∈ pathT (u, ū) and g ∈ pathTCL † (w, w̄) and the following holds under the error event T 6= TCL † : ( n† ∑