Suppose we are working with connected simple graphs.

We say the graph $G$ and $H$ are equivalent if for any spanning tree $T_G$ in $G$ there is an spanning tree $T_H$ in $H$ such that $T_G$ is isomorphic to $T_H$ and vice versa.

Can we say that if $G$ and $H$ are equivalent, then they are isomorphic?

The motivation of this question goes back to the reconstruction conjecture. I want to say that if $G$ and $H$ are equivalent and RC is true for one of these graphs, then it is true for other one.