  We present a quantitative basis-independent analysis of combinatory logic.
Using a general argument regarding plane binary trees with labelled leaves, we
generalise the results of David et al. and Bendkowski et al. to all
Turing-complete combinator bases proving, inter alia, that asymptotically
almost no combinator is strongly normalising nor typeable. We exploit the
structure of recently discovered normal-order reduction grammars showing that
for each positive $n$, the set of $\mathbf{S} \mathbf{K}$-combinators reducing
in $n$ normal-order reduction steps has positive asymptotic density in the set
of all combinators. Our approach is constructive, allowing us to systematically
find new asymptotically significant fractions of normalising combinators. We
show that the density of normalising combinators cannot be less than $34\%$,
improving the previously best lower bound of approximately $3\%$. Finally, we
present some super-computer experimental results, conjecturing that the density
of normalising combinators is close to $85\%$.
