What is the error when using Levenshtein distance on flattened binary trees?
This approximation is essentially arbitrarily bad.

Suppose you have a complete binary tree $T$ containing $2^k - 1$ nodes, of which the bottom $2^{k-1}$ are leaves.  Let $u$ and $v$ be the left and right child of the root, and $U$ and $V$ be the subtrees rooted at these vertices, respectively.  Let $w$ be the rightmost leaf in $U$.  To transform $T$ into a tree in which the locations of the leaf $w$ and the entire subtree $V$ have been swapped (so that the new tree has height $2k-1$) requires $2^{k-1}-1$ deletions of all nodes in $V$, the deletion of $w$, followed by the same number of insertions -- but the Levenshtein distance between the corresponding strings is just 2: delete the single list element corresponding to $w$ (from just before the half-way point of the string) and append it to the end.

(Well, possibly up to 2 tree-edits can be saved by turning pairs of insertions and deletions into substitutions.)