Computing the edit distance of two strings is one of the most basic problems in computer science and combinatorial optimization. Tree edit distance is a natural generalization of edit distance in which the task is to compute a measure of dissimilarity between two (unweighted) rooted trees with node labels. Perhaps the most notable recent application of tree edit distance is in NoSQL big databases, such as MongoDB, where each row of the database is a JSON document represented as a labeled rooted tree, and finding dissimilarity between two rows is a basic operation. Until recently, the fastest algorithm for tree edit distance ran in cubic time (Demaine, Mozes, Rossman, Weimann; TALG'10); however, Mao (FOCS'21) broke the cubic barrier for the tree edit distance problem using fast matrix multiplication. Given a parameter $k$ as an upper bound on the distance, an $O(n+k^2)$-time algorithm for edit distance has been known since the 1980s due to the works of Myers (Algorithmica'86) and Landau and Vishkin (JCSS'88). The existence of an $\tilde{O}(n+\mathrm{poly}(k))$-time algorithm for tree edit distance has been posed as an open question, e.g., by Akmal and Jin (ICALP'21), who gave a state-of-the-art $\tilde{O}(nk^2)$-time algorithm. In this paper, we answer this question positively.

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