It is a well-known elementary classical result in computability theory that there are computable infinite binary trees $T\subset 2^{<\omega}$ having no computable infinite branch. (One can build such a tree $T$ as follows: fix a computably inseparable pair of c.e. sets $A$ and $B$, and allow a binary string $t$ into the tree when the pattern of yes/no answers is consistent with being a separation of $A$ and $B$ in as much as these sets are revealed by stage $|t|$. In this way, incorrect guesses about the separation are eventually killed off, stuck in a finite dead part of the tree, and there can be no computable infinite branch because the sets have no computable separation.)


What I have need of is a strengthening of that classical result to the following:


**Question.** Are there two computable trees $T,S\subset 2^{\lt\omega}$ such that $T$ has an infinite branch not computing any infinite branch through $S$ and $S$ has a branch not computing any branch through $T$? 


That is, I want computable trees $S$ and $T$ such that there are infinite binary sequences $s$ and $t$, which are branches through $S$ and $T$, respectively, such that no infinite binary sequence that is Turing computable from $s$ is a branch through $T$ and no infinite binary sequence that is Turing computable from $t$ is a branch through $S$. (So in particular, neither $S$ nor $T$ can have any computable infinite branches.)



Are there two computable binary trees such that each has a branch not computing any branch through the other?
An earlier reference for a stronger result: Jockusch and Soare gave an example of a pair of $\Pi^0\_1$ classes such that any pair of elements consisting of one set from each class forms a minimal pair in the Turing degrees. Here is the bibliographic data from MathSciNet.



```
@article {MR0282834,
    AUTHOR = {Jockusch, Jr., Carl G. and Soare, Robert I.},
     TITLE = {A minimal pair of {$\Pi _{1}{}^{0}$} classes},
   JOURNAL = {J. Symbolic Logic},
  FJOURNAL = {The Journal of Symbolic Logic},
    VOLUME = {36},
      YEAR = {1971},
     PAGES = {66--78},
      ISSN = {0022-4812},
   MRCLASS = {02.70},
  MRNUMBER = {0282834 (44 \#68)},
MRREVIEWER = {R. L. Goodstein},
}   

```

