  We study certain generalizations of the classical Bernstein operators,
defined via increasing sequences of nodes. Such operators are required to fix
two functions, $f_0$ and $f_1$, such that $f_0 > 0$ and $f_1/ f_0$ is
increasing on an interval $[a,b]$. A characterization regarding when this can
be done is presented. From it we obtain, under rather general circumstances,
the following necessary condition for existence: if nodes are
non-{\guillemotleft}decreasing, then $(f_1/f_0)^\prime >0 $ on $(a,b)$, while
if nodes are strictly increasing, then $(f_1/f_0)^\prime >0 $ on $[a,b]$.
