  We introduce and study a new topology on trees, that we call the countably
coarse wedge topology. Such a topology is strictly finer than the coarse wedge
topology and it turns every chain complete, rooted tree into a
Fr\'echet--Urysohn, countably compact topological space. We show the r\^{o}le
of such topology in the theory of weakly Corson and weakly Valdivia compacta.
In particular, we give the first example of a compact space $T$ whose every
closed subspace is weakly Valdivia, yet $T$ is not weakly Corson. This answers
a question due to Ond\v{r}ej Kalenda.
