Let's denote by the number of labeled ( label set ) trees - with n vertices- such that the degree of vertex is .
Then we have the generating function:
Now if what we want is the sum of the entries where the corresponding exponent of is 1.
But of course if we require the exponent of to be one we have the generating function: since we may never choose in one of the we had in .
Thus the number of trees such that vertex u is a leaf is:
And the total number of labelled trees is .
For further details on this read here.