Suppose you have a labeled tree $T$ on vertices $V=\lbrace 1,\ldots,n\rbrace$ that is drawn uniformly at random from the set of all $n^{n-2}$ such trees. I am seeking an $f$ satisfying the following desiderata:

D1. $f(T)$ is a (random) tree on the vertex set $V'=\lbrace 1,\ldots,n+1\rbrace$;

D2. the distribution of $f(T)$ is uniform over the set of all $(n+1)^{n-1}$ labeled trees on $V'$;

D3. $f$ is a simple graph-theoretic recipe.

This $f$ can be a random function. That is, it can flip coins in deciding what to do with $T$: for instance, removing an edge uniformly at random.

One obvious way to satisfy D1 and D2 is:

convert $T$ to its Prüfer sequence $s$;

independently, with probability $1/(n+1)$, change each entry of $s$ to $n+1$ (otherwise leave it fixed);

then add a random number to the end of $s$, drawn uniformly from $V'$;

let $f(T)$ be the tree corresponding to the new sequence.

But what this procedure "really does" to $T$ seems not so easy to describe in graph-theoretic terms. I am looking for a recipe that satisfies D1-3 by manipulating the graph "directly" (i.e., adding and removing edges) without opaque steps like (1) and (4) in the above procedure.