If T is a tree having one vertex of degree i for each i = 2,...,k, then you can get then next one by taking one of the end-vertices and adding k new edges to k new end-vertices.
Here's an illustration for k=2,3 and 4. Each time, you take the vertex at the right-hand end of the diagram and add new edges to it to get the next diagram.
There is only one vertex of degree i for each i>1, hence k-1 such vertices as k goes from 2 to k.
Add two new edges to get:
Add three new edges to get:
Now add four new edges to the rightmost vertex...