$$2\left (\frac {t-1} 2 +5\right )-1=\frac{2t-2}2+10-1=t+8=m$$ pairwise non-isomorphic trees. Since the proof of Theorem \[lem:CPS-Existence\] generalizes to a set of $m$ intermezzo trees, the corollary now follows for both cases.

Bounding the number of cherries {#sec:cherries}
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The main result of this section is the following theorem.

\[t:cherry\] Let ${{\mathcal P}}=\{{{\mathcal T}}_1,{{\mathcal T}}_2,\ldots,{{\mathcal T}}_m\}$ be a collection of rooted binary phylogenetic $X$-trees. Let $c$ be the maximum element in $\{c_{{{\mathcal T}}_1},c_{{{\mathcal T}}_2},\ldots,c_{{{\mathcal T}}_m}\}$. Then solving [CPS-Existence]{} for ${{\mathcal P}}$ takes time $$O\left (|X|^{m(4c-2)+1} + \sum_{i=1}^m f_i(|X|, c_{{{\mathcal T}}_i})\right ),$$ where $f_i(|X|, c_{{{\mathcal T}}_i}) \in |X|^{O(c_{{{\mathcal T}}_i})}$. In particular, the running time is polynomial in $|X|$ if $c$ and $m$ are constant.

Let ${{\mathcal T}}$ be a rooted binary phylogenetic $X$-tree. We denote by ${{\mathcal C}}({{\mathcal T}})$ the recursively defined set of trees that contains ${{\mathcal T}}$ and $\emptyset$, and that satisfies the following property.

[**(P)**]{} If a tree ${{\mathcal T}}'$ is in ${{\mathcal C}}({{\mathcal T}})$ and $\{a,b\}$ is a cherry in ${{\mathcal T}}'$, then ${{\mathcal T}}'[-a]$ and ${{\mathcal T}}'[-b]$ are also contained in ${{\mathcal C}}({{\mathcal T}})$.

We refer to ${{\mathcal C}}({{\mathcal T}})$ as the [*set of cherry-picked trees*]{} of ${{\mathcal T}}$. Intuitively, ${{\mathcal C}}({{\mathcal T}})$ contains each tree that can 