  Let $G = (V, E)$ be an undirected connected simple graph on $n$ vertices. A
cut-equivalent tree of $G$ is an edge-weighted tree on the same vertex set $V$,
such that for any pair of vertices $s, t\in V$, the minimum $(s, t)$-cut in the
tree is also a minimum $(s, t)$-cut in $G$, and these two cuts have the same
cut value. In a recent paper [Abboud, Krauthgamer and Trabelsi, 2021], the
authors propose the first subcubic time algorithm for constructing a
cut-equivalent tree. More specifically, their algorithm has
$\widetilde{O}(n^{2.5})$ running time.
  In this paper, we improve the running time to $\hat{O}(n^2)$ if almost-linear
time max-flow algorithms exist. Also, using the currently fastest max-flow
algorithm by [van den Brand et al, 2021], our algorithm runs in time
$\widetilde{O}(n^{17/8})$.
