Do you know how many nodes are in each node's child subtrees? If you do, you can just decide that you want, say, the $k$-th node from the left and then descend the tree to find that node:
Let $n$ be the total number of nodes in the tree. Choose $k$ to be a random integer between $0$ and $n-1$ inclusive. Let $A$ initially be the root node of the tree.
Let $m$ be the number of nodes in the left subtree of $A$. (If $A$ is a leaf or has only right children, let $m = 0$.)
If $k = m$, choose $A$ as the node we want and stop.
Otherwise, if $k < m$, replace $A$ with its left child node and repeat from step 2.
Otherwise (i.e. if $k > m$), subtract $m+1$ from $k$, replace $A$ with its right child node and repeat from step 2.
This algorithm is much more efficient than traversing the entire tree; its running time is bounded by the depth of the tree, which for (approximately) balanced trees is proportional to the logarithm of the total number of nodes.