Tree Representations of Contextual Equivalence

For a set $\mathcal{P}$ of lambda terms, define $M\sim_{\mathcal{P}}N$ when, for every one-hole context $C[\,\cdot\,]$, $C[M]$ beta-reduces to a member of $\mathcal{P}$ if and only if $C[N]$ does. Find a tree representation whose equality is exactly $\sim_{\mathcal{P}}$ for each of the following choices:

1. $\mathcal{P}=\mathcal{W}$, the weak-head normal forms;

2. $\mathcal{P}=\mathcal{T}$, the top normal forms;

3. $\mathcal{P}=\mathcal{SA}$, the strongly active terms;

4. $\mathcal{P}=\mathcal{SA}_X$, the strongly active terms depending on a fixed set $X$ of lambda terms;

5. $\mathcal{P}=\mathcal{HA}$, the head-active terms.