Tree edit distance (TED), the analog of string edit distance (ED) for hierarchical data, is the minimum number of node insertions, deletions, and substitutions transforming one labeled rooted ordered tree into another. We narrow polynomial gaps between TED and ED in near-linear-time approximation, sublinear algorithms, and dynamic algorithms. We give an $n^{o(1)}$-approximation in $O(n)$ time and, for every $0<δ<1$, a $(\log n)^{O(1/δ)}$-approximation in $O(n^{1+δ})$ time. This improves the previous near-linear-time $O(\sqrt n)$-approximation [Boroujeni et al., STOC'19] and brings TED to parity with ED in this regime up to polylogarithmic factors. With query access to the balanced-parentheses (Euler-tour) representation of the input trees, we give the first sublinear-time TED algorithm: for every $τ\in[1,n]$, it computes a $τn^{o(1)}$-approximation of $1+\mathrm{ted}$ in $O(n/τ)$ time. For dynamic TED, we give an exact algorithm with amortized update time $\tilde{O}(k^7)$ and a $(1+\varepsilon)$-approximation with amortized update time $\tilde{O}_{\varepsilon}(k)$, where $k$ caps the distance that must be reported. The exact algorithm is the first to beat recomputation in a nontrivial parameter regime. The approximation improves prior dynamic guarantees [Das et al., FOCS'25] and cannot be significantly improved in any parameter regime without a breakthrough for static $(1+\varepsilon)$-approximation of ED. All results use dynamic programming on a new hierarchical tree decomposition, analogous to partitioning strings into blocks of geometrically decreasing sizes. It combines structural properties previously achieved only in isolation, providing a framework for transferring algorithmic ideas from ED to TED, with additional ingredients addressing the obstacles in each computational setting.
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