Suppose I give you $n$ vertices, I tell you that these vertices are connected by edges to form a tree, and I tell you that the degree of the vertices are $d_1,\ldots,d_n$.
Does that determine the tree (i.e., the edges of the tree) uniquely?
Suppose I give you $n$ vertices, I tell you that these vertices are connected by edges to form a tree, and I tell you that the degree of the vertices are $d_1,\ldots,d_n$.
Does that determine the tree (i.e., the edges of the tree) uniquely?
No it does not. Consider two trees each with $7$ vertices and degrees $1, 1, 1, 2, 2, 2, 3$.