In this paper, we introduce and study a new multivalued mapping in $mathbb{R}$-trees, called emph{$k$-strictly pseudononspreading}. We also introduce a new two-step iterative process for two $k$-strictly pseudononspreading
multivalued mappings in $mathbb{R}$-trees. Strong convergence theorems of the proposed iteration to a common fixed point of two $k$-strictly pseudononspreading multivalued mappings in $mathbb{R}$-trees are established. Our results improve and extend the corresponding results existing in the literature.

Keywords: fixed point, $mathbb{R}$-tree, strictly pseudononspreading mapping, convergence theorems