  We find sufficient conditions under which the product of spaces that have a
$\pi$-tree also has a $\pi$-tree. These conditions give new examples of spaces
with a $\pi$-tree: every at most countable power of the Sorgenfrey line and
every at most countable power of the irrational Sorgenfrey line has a
$\pi\!$-tree. Also we show that if a space has a $\pi$-tree, then its product
with the Baire space, with the Sorgenfrey line, and with the countable power of
the Sorgenfrey line also has a $\pi$-tree.
