Suppose for a graph $G=(V,E)$ and a spanning tree T of G, $\Delta(T)$ is the largest degree of a vertex in T, and let $\Delta^*$ be the smallest such quantity over all spanning trees of $G$.
We have the following local search procedure which can changes spanning tree $T$ into a different spanning tree $T'$: We find an edge $e$ not in $T$ and add it to $T$. This results in a cycle $C$ - call its vertices vertices $V(C)$. We then delete an edge in $C$ incident to a vertex in $V(C)$ with highest degree.
My question is this: if $\Delta(T) > \Delta^*$, can we always find an edge $e$ to add, such that the maximum degree of vertices in $V(C)$ is strictly less in $T'$ than it is in $T$?