Interval edge-colorings of trees with restrictions on the edges

An edge-coloring of a graph $G$ with consecutive integers $c_1,\ldots,c_t$ is called an interval $t$-coloring, if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable, if it has an interval $t$-coloring for some positive integer $t$. In this paper, we consider the case, where there are restrictions on the edges of the tree and provide a polynomial algorithm for checking interval colorability that satisfies those restrictions.