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For the ordered set $[n]$ of $n$ elements, we consider the class $\Bscr_n$ ofbases $B$ of tropical Pl\"ucker functions on $2^{[n]}$ such that $B$ can beobtained by a series of mutations (flips) from the basis formed by theintervals in $[n]$. We show that these bases are representable by specialwiring diagrams and by certain arrangements generalizing rhombus tilings on the$n$-zonogon. Based on the generalized tiling representation, we then prove thateach weakly separated set-system in $2^{[n]}$ having maximum possible sizebelongs to $\Bscr_n$, thus answering affirmatively a conjecture due to Leclercand Zelevinsky. We also prove an analogous result for a hyper-simplex$\Delta_n^m=\{S\subseteq[n]\colon |S|=m\}$.

Pl\"ucker environments, wiring and tiling diagrams, and weakly separated set-systems