  We investigate the question whether a scattered compact topological space $K$
such that $C(K)$ has a norming Markushevich basis (M-basis, for short) must be
Eberlein. This question originates from the recent solution, due to H\'ajek,
Todor\v{c}evi\'c, and the authors, to an open problem from the Nineties, due to
Godefroy. Our prime tool consists in proving that $C([0,\omega_1])$ does not
embed in a Banach space with a norming M-basis, thereby generalising a result
due to Alexandrov and Plichko. Subsequently, we give sufficient conditions on a
compact $K$ for $C(K)$ not to embed in a Banach space with a norming M-basis.
Examples of such conditions are that $K$ is a $0$-dimensional compact space
with a P-point, or a compact tree of height at least $\omega_1 +1$. In
particular, this allows us to answer the said question in the case when $K$ is
a tree and to obtain a rather general result for Valdivia compacta. Finally, we
give some structural results for scattered compact trees; in particular, we
prove that scattered trees of height less than $\omega_2$ are Valdivia.
