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 endvertices and adding k new edges to k new endvertices.
Here's an illustration for k=2,3 and 4. Each time, you take the vertex at the righthand end of the diagram and add new edges to it to get the next diagram.
Code:
o——o——o
Add two new edges to get:
o——o——o——o

o
Add three new edges to get:
o

o——o——o——o——o
 
o o
Now add four new edges to the rightmost vertex...
There is only one vertex of degree i for each i>1, hence k1 such vertices as k goes from 2 to k.