on the underlying trees.

[**Subtrees.**]{} Now, let ${{\mathcal T}}$ be a rooted binary phylogenetic $X$-tree, and let $X'=\{x_1,x_2,\ldots,x_k\}$ be a subset of $X$. The minimal rooted subtree of ${{\mathcal T}}$ that connects all vertices in $X'$ is denoted by ${{\mathcal T}}(X')$. Furthermore, the rooted binary phylogenetic tree obtained from ${{\mathcal T}}(X')$ by contracting all non-root degree-$2$ vertices is the [*restriction of ${{\mathcal T}}$ to $X'$*]{} and is denoted by ${{\mathcal T}}|X'$. We also write ${{\mathcal T}}[-x_1,x_2,\ldots,x_k]$ or ${{\mathcal T}}[-X']$ for short to denote ${{\mathcal T}}|(X-X')$. For a set ${{\mathcal P}}=\{{{\mathcal T}}_1,{{\mathcal T}}_2,\ldots,{{\mathcal T}}_m\}$ of rooted binary phylogenetic $X$-trees, we write ${{\mathcal P}}|X'$ (resp. ${{\mathcal P}}[-X']$) when referring to the set $\{{{\mathcal T}}_1|X',{{\mathcal T}}_2|X',\ldots,{{\mathcal T}}_m|X'\}$ (resp. $\{{{\mathcal T}}_1[-X'],{{\mathcal T}}_2[-X'],\ldots,{{\mathcal T}}_m[-X']\}$). Lastly, a rooted binary phylogenetic tree is [*pendant*]{} in ${{\mathcal T}}$ if it can be detached from ${{\mathcal T}}$ by deleting a single edge.

[**Cherry-picking sequences.**]{} Let ${{\mathcal P}}$ be a set of rooted binary phylogenetic $X$-trees with $|X|=n$. We say that an ordering of the elements in $X$, say $(x_1,x_2,\ldots,x_n)$, is a [*cherry-picking sequence*]{} for ${{\mathcal P}}$ precisely if each $x_i$ with $i\in\{1,2,\ldots,n-1\}$ labels a leaf of a cherry in each tree that is contained in ${{\mathcal P}}[-x_1,x_2,\ldots,x_{i-1}]$. Clearly, if 