Pseudotrees and equivalent norms in the continuous. Functions spaces

A class of the pseudotrees is considered. We construct locally compact extension of a pseudotree, which also has the structure of a pseudotree. We prove that the space $C_0(T)$ of all continuous functions on a locally compact pseudotree $T$ admits a locally uniform rotund (LUR) renorming if the related space $C_0(P)$ admits such norm for every subtree $P$ of $T$ and an initial segments of $T$ are separable.