The topos of trees is the category of presheaves over the ordinal of natural numbers, $\omega$. More properly, this is the category of trees of height bounded by $\omega$, in that every path from the root has length $\omega$ or less (when considered as ordinals). These trees can, however, be arbitrarily branching at every level.

An object is then a family of sets $X_i$ for each $i \in \omega$ with restriction functions $X_i \leftarrow X_{i+1}$. We can visualize this as a (potentially infinite) tree (really, forest) where an element of any $X_i$ is a node of the tree, and the restriction functions map each node to its parent node.