If the nodes add a fixed amount of new nodes at each level, then the number of infinite paths seems to be countable. -
It does not seem so for me. Even if the number of nodes $k_n$ on level $n$ satisfies $k_n\leqslant k_{n+1}\leqslant k_n+1$, the number of infinite paths may have cardinality continuum. For constructing an example take an infinite binary tree and replace each edge to a path of certain length, choosing lengths so that all branchings occur at different levels. This is possible, just choose the lengths successively.