The following is from a paper by Joel Spencer, called Enumerating Graphs and Brownian Motion.
Let us look at a tree $T$ on vertex set $\{1, \ldots, n\}$ from a computer science perspective. We'll take $n=7$ and the tree with edges 13,15,16,27,34,37 as an example throughout. Start with vertex 1 and find the tree through a breadth-first search (BFS), all lists being in numerical order. To do this we have a queue $Q$, that is initially (1). At each stage we take the vertex $x$ at the head of the queue, remove $x$ from the queue, and add all "new" neighbors of $x$ to the queue. We'll call that operation "processing $x$ " and say that $x$ finds $y$ if $y$ is a new vertex adjacent to $x$. In our example, 1 finds $3,5,6,$ then 3 finds $4,7,$ then 5 finds nothing, 6 finds nothing, 4 finds nothing, 7 finds $2,$ and finally 2 finds nothing. After 2 is processed, the queue becomes empty and the BFS ends.
Let $x_{t}$ be the number of vertices found by the $t^{\text {th }}$ vertex processed (not vertex number $t$ ), which, in our example, is 3,2,0,0,0,1,0 . Let $q_{t}$ be the size of the queue after the $t^{\text {th }}$ vertex is processed. Then $q_{0}=1$ and $q_{t}= q_{t-1}+x_{t}-1$, which, in our example, is $1,3,4,3,2,1,1,0$ . The necessary and sufficient conditions on the sequence $q_{t}$ are $$ q_{n}=0 \quad \text { and } \quad q_{i}>0 \quad \text { for } i<n $$ Now note that there are precisely $(n-1) ! / \prod x_{i} !$ trees with a given sequence $x_{t}$. Indeed, any partition of the vertex set excluding source vertex 1 into sets of sizes $x_{1}, \ldots, x_{n}$ gives a unique such tree $T$.
Random Trees
Let $X_{1}, \ldots, X_{n}$ be independent random variables, each Poisson with mean $1$ . That is, $\operatorname{Pr}\left[X_{i}=x\right]=e^{-1} / x !$ . Define $Q_{0}=1$ and $Q_{t}=Q_{t-1}+X_{t}-1$ . Condition on $$ Q_{n}=0 \quad \text { and } \quad Q_{i}>0 \quad \text { for } i<n $$ The probability of having values $x_{1}, \ldots, x_{n}$ is then proportional to $1 / \prod x_{i} !$ and the support for the $X_{t}$ is that for trees so that the conditional distribution is precisely that for the $\left(x_{1}, \ldots, x_{n}\right)$ when a tree $T$ is selected from all trees on $n$ vertices with a uniform distribution.
I do not understand what is meant by "the support for the $X_{t}$ is that for trees so that the conditional distribution is precisely that for the $\left(x_{1}, \ldots, x_{n}\right)$ when a tree $T$ is selected from all trees on $n$ vertices with a uniform distribution." More specifically the part about support for $X_t$.
Can anyone clarify this for me?
The support of a probability measure is the set where it is non-zero (https://en.wikipedia.org/wiki/Support_(measure_theory)).
What he says is that, once you condition on $Q_n = 0, Q_i > 0$ then your sequence of $(X_1, ..., X_n)$ will look like the sequences he described before. Moreover, he claims that this is the same distribution as if you had taken a uniformly random tree and taken its vertex discovery sequence. I guess he proves these facts just afterwards in the paper.