The present work consists of 3 sections. In section 1 we have Theorem 1:1 which gives an sufficient condition to exhibit a kind of antichains in pseudotrees. In section 2 the problem of attainability of $p_sE$ is examined: since simple examples show that even in well-founded sets $W$ the number $p_s W$ might be unattained one examines the case of $p_sT$ for trees; we prove the main Theorem 2:4 and formulate ATH (Antichain Tree Hypothesis) in 2:7 and prove that ATH is implied by the RH (Ramification Hypothesis) (v. 2:8 Theorem). We stress the fact how limit regular cardinals occur in considerations in section 2. Section 3 examines $p_sT^n$ for squares, cubes and hypercubes of trees it is proved that for any index set $I$ of cardinality $>1$ the cardinal ordering of the hypercube $T^I$ is such that the number $p_sT^I$ is attained. One has the beautiful result 3:5.