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I found this paper about "Almost all trees are cospectral" as the number of nodes go asymptotically large.

But it is pretty hard to do experiments to find cospectral but non-isomorphic trees for small number of nodes. Are there simple examples with very few nodes?

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One of small examples is on Fig. 2 of the paper “Topological twin graphs II. Isospectral polyhedral graphs with nine and ten vertices” by Haruo Hosoya, Kyoko Ohta, and Masaki Satomi.

PS. This my answer contains few references about cospectral graphs.

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