In this paper, we study $\lambda$-representability in the simple type
system of functions over free structures as well as those over the
sets of first-order terms.  We first prove a characterization theorem
for $\lambda$-representable functions over the sets of first-order
terms.  Then based on the result we obtain two characterizations of
$\lambda$-representable functions over free structures, one of which
is a byproduct of our new proof of a Zaionc result.  The central
notion of our characterizations is a variant of primitive recursion
scheme which is similar but different from Zaionc limited primitive
recursion.