Tree $T$ and Lambert $W$

The paper discusses the two cognate functions Tree
$T$
and Lambert
$W$
. The definition of the Lambert
$W$
function in the complex plane, specifically the branch structure, has been the subject of varying opinions from the time
$W$
was named. These opinions are reviewed. Until now, the cognate functions have been regarded as simple reflections of each other. Here I propose that the two functions be given independent branch structures, thus allowing users of computer algebra systems access to whichever structure is convenient for them.