Constructing trees with the same degree sequences I've got this problem.

Let $G$, $H$ be the trees (simple graphs) with the same degree sequences. Is it true that there always be vertices $q\in V(G)$ and $q′\in V(H)$ such that $(q,p)\in E(G)$ and $(q′,p′)\in E(H)$ for some endvertices $p\in V(G)$ and $p\in V(H)$, and $d(q)=d(q′)$?

$d(q)$ - degree of the vertex $q$.

I haven't found counterexample for trees up to $8$ vertices, and it's seems impossible to me.

Have you references for some results concerned with trees with the same degree sequences?