\:Record~$v$ in the sequence.
\:For each son~$w$ of~$v$:
\::Call $\<ET>(w)$.
-\::Record~$w$.
+\::Record~$v$.
\endalgo
\>A~single tree can have multiple ET-sequences, corresponding to different orders in which the
sons can be enumerated in step~2.
\em{Changing the root} of the tree~$T$ from~$v$ to~$w$ changes its ET-sequence from $vAwBwCv$ to $wBwCvAw$.
If $w$~was a~leaf, the sequence changes from $vAwCv$ to $wCvAw$. If $vw$ was the only edge of~$T$,
-the sequence $vw$ becomes $wv$. Note that this works regardless of the possible presence of~$w$ inside~$B$.
+the sequence $vwv$ becomes $wvw$. Note that this works regardless of the possible presence of~$w$ inside~$B$.
\em{Joining} the roots of two trees by a~new edge makes their ET-sequences $vAv$ and~$wBw$
combine to $vAvwBwv$. Again, we have to handle the cases when $v$ or~$w$ has degree~1 separately: