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. Moreover we prove that the space of continuous functions on an arbitrary tree with height less than ω 1 · ω 0 is Plichko.MSC: 46B26, 46A50, 54D30, 06A06.

The aim of this note is to characterize trees, endowed with coarse wedge topology, that have a retractional skeleton. We use this characterization to provide new examples of non-commutative Valdivia compact spaces that are not Valdivia.MSC: 54D30, 54C15, 54G20, 54F05

The new concepts are introduced of almost overcomplete sequence in a Banach space and almost overtotal sequence in a dual space. We prove that any of such sequences is relatively norm-compact and we obtain several applications of this fact