  In the present paper we investigate the class of compact trees, endowed with
the coarse wedge topology, in the area of non-separable Banach spaces. We
describe Valdivia compact trees in terms of inner structures and we
characterize the space of continuous functions on them. We prove that all
compact trees have the property $(M)$. Finally we prove that the space of
continuous functions on an arbitrary tree with height less than
$\omega_1\cdot\omega_0$ is Plichko.
