[**Phylogenetic trees.**]{} A [*rooted binary phylogenetic $X$-tree*]{} ${{\mathcal T}}$ is a rooted tree with leaf set $X$ and, apart from the root which has degree two, all interior vertices have degree three. Furthermore, a pair of leaves $\{a,b\}$ of ${{\mathcal T}}$ is called a [*cherry*]{} if $a$ and $b$ are leaves that are adjacent to a common vertex. Note that every rooted binary phylogenetic tree has at least one cherry. We denote by $c_{{\mathcal T}}$ the number of cherries in ${{\mathcal T}}$. We now turn to a rooted binary phylogenetic tree with exactly one cherry. More precisely, we call ${{\mathcal T}}$ a [*caterpillar*]{} if $|X|=n\geq 2$ and the elements in $X$ can be ordered, say $x_1, x_2, \ldots, x_n$, so that $\{x_1, x_2\}$ is a cherry and, if $p_i$ denotes the parent of $x_i$, then, for all $i\in \{3, 4, \ldots, n\}$, we have $(p_i, p_{i-1})$ as an edge in ${{\mathcal T}}$, in which case we denote the caterpillar by $(x_1, x_2, \ldots, x_n)$. To illustrate, Figure \[fig:caterpillar\] shows the caterpillar $(D_1,D_2,\ldots,D_{|A'|})$ with cherry $\{D_1,D_2\}$. Two rooted binary phylogenetic $X$-trees ${{\mathcal T}}$ and ${{\mathcal T}}'$ are said to be [*isomorphic*]{} if the identity map on $X$ induces a graph isomorphism 