  We consider the positive part $U^+_q$ of the quantized enveloping algebra
$U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentation
involving two generators and two relations, called the $q$-Serre relations.
There is a PBW basis for $U^+_q$ due to Damiani, and a PBW basis for $U^+_q$
due to Beck. In 2019 we used Catalan words and a $q$-shuffle algebra to express
the Damiani PBW basis in closed form. In this paper we use a similar approach
to express the Beck PBW basis in closed form. We also consider how the Damiani
PBW basis and the Beck PBW basis are related to the alternating PBW basis for
$U^+_q$.
