You can build a perfect tree where the branching happens always and only at certain specified levels.

There is an antichain of $2^n $ many finite strings $\sigma_{i, n} $ of length $2^n $.

Consider sequences $\tau =\lim_{n}\tau_n$ where
$$\tau_n = \tau_{n-1}\sigma_{[\tau_{n-1}]\cdot 2+i_n, n} $$
 (concatenation denoted by juxtaposition here) where the $[\sigma]$th string of the same length as $\sigma$ is $\sigma $. There are continuum many choices of $ i_n \in \{0,1\}$ in an infinite sequence $\tau$ and these $\tau $ form an antichain.