A subset $A$ of a poset $(P,<_{P})$ is an antichain if no two elements are comparable. That is, if $a,b\in A$ then $a\nless_{P}b$ and $b\nless_{P}a$.

A maximal antichain of $T$ is one which is maximal.

In particular, if $(P,<_{P})$ is a tree then the maximal antichains are exactly those antichains which intersect every branch, and if the tree is splitting then every level is a maximal antichain.