Hardness of Dynamic Tree Edit Distance and Friends

String Edit Distance is a more-than-classical problem whose behavior in the dynamic setting, where the strings are updated over time, is well understood. A single character substitution, insertion, or deletion can be handled in $\tilde{\mathcal{O}}(n \cdot \min(\sqrt{n},w))$ time [Charalampopoulos, Kociumaka, Mozes, CPM 2020], where $w$ is the maximum operation weight. This bound is optimal [Cassis, Kociumaka, Wellnitz, FOCS 2023] and provides a substantial improvement over the static $\mathcal{O}(n^2)$ algorithm when few characters of the input string are updated. In contrast, for inherently related problems such as Tree Edit Distance, Dyck Edit Distance, and RNA Folding, it has remained unknown whether it is possible to devise dynamic algorithms with an advantage over the static algorithm. In this paper, we resolve this question by showing that (weighted) Tree Edit Distance, Dyck Edit Distance, and RNA Folding admit no dynamic speedup: under well-known fine-grained assumptions we show that the best possible algorithm recomputes the solution from scratch after each update. Furthermore, we prove a quadratic per-update lower bound for unweighted Tree Edit Distance under the $k$-Clique Conjecture. This provides the first separation between dynamic unweighted String Edit Distance and unweighted Tree Edit Distance, problems whose relative difficulty in the static setting is still open.