Conway's well known 'Look and Say' (sometimes called an audioactive sequence) sequence is the following: $1,11,21,1211,111221,312211,...$
Another example of a 'Look and Say' sequence is $55555,55,25,1215,11121115,31123115,....$
More formally, a 'Look and Say' sequence is constructed by starting with a number $l_0 \in \mathbb{N}$, and the following terms are given by Conway's audioactive operator $l_n = JHC(l_{n-1})$, where
$JHC(a_1^{n_1}a_2^{n_2}a_3^{n_3}...a_k^{n_k}) = n_1a_1n_2a_2n_3a_3 ... n_ka_k$
where $a^m$ is a short hand notation for $a$ repeated $m$ times.
What I'd like to know is that if you took each term to instead be a Prüfer Code (provided it's not a degenerate code) you produce a sequence of trees. Has there been an analysis done on these kinds of sequences of trees?