Homomorphism Counts to Trees

We construct a pair of non-isomorphic, bipartite graphs which are not distinguished by counting the number of homomorphisms to any tree. This answers a question raised by Atserias et al. (LICS 2021). In order to establish the construction, we analyse the equivalence relations induced by counting homomorphisms to trees of diameter two and three and obtain necessary and sufficient conditions for two graphs to be equivalent. We show that three is the optimal diameter for our construction.