You can build a perfect tree where the branching happens always and only at certain specified levels. Then you put an antichain of finite strings of the same length across each level. Voila.

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.