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? 



Trees with the same degree sequences
Consider two trees $G$ and $H$ with 14 vertices. Both will have degree sequence $(2,0,6,6)$ i.e. having two vertices of degree 4. $G$ will have the two 4-vertices connected to 3 leaves each and with a 6 vertex long chain between them. $H$ will have the two 4-vertices connected with a single edge. In addition, each will have three 2 vertex long chains connected to them (one 2-vertex connected to a leaf).


Finally, each leaf of $G$ is connected to a 4 vertex and each leaf of $H$ is connected to a 2 vertex.


A picture would do the trick better.


