I would like to know if there exists a result saying that for a fixed undirected rooted Eulerian graph, up to some permutation, along any Eulerian cycle, there exists a unique sequence of degrees, where the degree of a vertex along an Eulerian path is (not the usual degree but) the number of neighboor vertices such that the path may be extended to an Eulerian cycle.

As an example, if we consider the graph composed of two vertices (one being the root) linked by $n$ parallel lines of $m_i$ edges each, then up to some permutation the only sequance of degrees along an Eulerian cycle is $(n,n-1,\dots,2,1,\dots,1)$ with $1$ appearing $\sum_{i=1}^n(m_i-1)+1$ times at the end.