How many cherry-picking sequences are needed to reduce all subtrees of a phylogenetic tree?

Abstract: Phylogenetic networks are graphs that represent the evolutionary history of species. Recently, the class of orchard phylogenetic networks, which can be reduced by so-called cherry-picking sequences, has gained attention for its computational and biological aspects. In this paper, we study a fundamental question on orchards and their cherry-picking sequences by considering the CoveringNumber problem: given an orchard network $N$, how many cherry-picking sequences are needed to reduce all subnetworks of $N$? We initiate this study by considering the problem for trees. We then show that the covering number can be computed for binary trees recursively using a similar but more fine-grained notion of survival covering number. We also give a recursive formula for the survival covering number of non-binary trees. However, computing the covering number for non-binary trees appears to be considerably more challenging. For this case, we show that the covering number of star trees (whose root is adjacent to all leaves) is equivalent to the so-called SubsetConnectivity problem, which we introduce in this paper. Finally, we show that if there is no restriction on the sequence length, a single sequence of minimum length $\binom{n}{2}$ suffices to reduce all subtrees of a tree on $n$ leaves.

• The paper derives a constructive recursion for binary trees, using survival covering numbers to determine optimal covers through products and sums of child-subtree values.
• For non-binary trees, the paper computes the survival covering number via binary refinements and caterpillarizations, while reducing unrestricted star-tree coverage to SubsetConnectivity.
• A single sequence of exactly \(\binom{n}{2}\) leaf-pair operations reduces every subtree, but covering numbers for general non-binary trees and some networks remain open or can be infinite.

This paper studies the CoveringNumber problem for phylogenetic trees: given a tree $T$ on $n$ leaves, what is the minimum number of cherry-picking sequences of $T$ needed to reduce every subtree of $T$? The authors, Kollmann, Murakami, and Suzuki (2608.19916), resolve this problem completely for binary trees via a recursive formula built on a finer-grained "survival" variant of the covering number, extend the survival variant to non-binary trees, and show that the general non-binary case is equivalent — even for star trees — to a new combinatorial optimization problem they call SubsetConnectivity. They also prove that a single sequence of length exactly $\binom{n}{2}$ suffices to reduce all subtrees when sequence length is unrestricted.

Background and motivation

Cherry-picking sequences are ordered pairs of leaves that iteratively reduce cherries (pairs of leaves sharing a parent) in a phylogenetic tree or network. They underpin polynomial-time algorithms for TreeContainment on tree-child and orchard networks: a tree-child network contains a tree on the same leaf set if and only if any tree-child sequence reduces that tree. When leaf sets differ, however, TreeContainment becomes NP-hard even for tree-child networks, motivating the question of how many sequences are needed to cover all subtrees. Dee previously introduced CoveringNumber for trees and solved it only for caterpillars and double caterpillars; this paper is the first treatment of general trees.

The key structural notion is the survival covering number $c'(T)$: a set of sequences survivably covers $\mathcal{T}_T$ if every subtree is not merely reduced but reduced with its last remaining leaf being the surviving leaf of the sequence with respect to some pendant subtree. This constraint ensures subtrees can be propagated upward and combined with other subtrees during reduction.

Covering number of binary trees

The central result is a closed recursion. Let $T$ be binary with root $\rho$, children $a,b$, and $n$0. Then:

$n$1

The proof proceeds in three stages. First, two lemmas construct pairwise (survivably) irreducible subtree sets: subtrees on disjoint leaf sets are always pairwise survivably irreducible, and irreducibility is preserved under concatenation with disjoint leaf sets outside a vertex's clade. Second, matching lower and upper bounds are established for $n$2: the lower bound comes from a maximum pairwise survivably irreducible set $n$3 satisfying $n$4, while the upper bound is constructive — an explicit matrix $n$5 of combined sequences $n$6 built from optimal survival covers of the two child subtrees, using arbitrary orderings of their leaves to guarantee correct surviving-leaf alignment. Since $n$7 by induction, both bounds coincide, yielding the recursion for $n$8 as well as for $n$9. All proofs are constructive, so optimal covering sets can be assembled bottom-up; computation is linear time in the input size.

Survival covering number for non-binary trees

For non-binary trees, the authors generalize via binary refinements and caterpillarization. A caterpillarization of a non-binary tree converts each high-degree vertex into a chain of binary vertices ordered by increasing subtree size. Two transfer lemmas show that restricting a binary refinement to a subset of leaves yields a binary refinement of the restricted original tree, and that any sequence reducing a refinement also reduces the original. Consequently, a survival covering set of any binary refinement covers the original non-binary tree.

The main theorem states that for any non-binary tree $T$0 with root children partitioned into non-leaf set $T$1 and leaf set $T$2:

$T$3

and this value equals the survival covering number of any caterpillarization of $T$4. Notably, the choice of ordering within the caterpillarization does not affect the value, though it may affect which optimal cover is produced. The proof is by induction on tree height, combining the binary recursion applied to the caterpillarization with the lower bound from pairwise survivably irreducible sets.

Star trees and SubsetConnectivity

The unrestricted covering number $T$5 for non-binary trees resists the techniques above. The authors isolate the difficulty by proving an equivalence for star phylogenetic trees $T$6, where every leaf is adjacent to the root: $T$7 is a YES-instance of CoveringNumber if and only if $T$8 is a YES-instance of the newly defined SubsetConnectivity problem, which asks whether there exist $T$9 unrooted labeled trees on $T$0 vertices such that every subset $T$1 induces a connected subgraph in at least one of them.

The equivalence is bi-constructive: a cherry-picking sequence maps to a tree whose edges are its pairs (acyclicity follows from the first-coordinate-uniqueness property of sequences), and conversely a complete family of unrooted trees yields sequences by repeatedly peeling degree-one vertices. For SubsetConnectivity itself, the paper establishes that $T$2, and that if $T$3 then no minimum complete family for $T$4 contains a star graph. Computationally, exact search confirms $T$5 for $T$6; whether this pattern persists for larger $T$7 remains open.

Reducing all subtrees with one long sequence

When sequences are allowed to exceed length $T$8, the picture simplifies dramatically. Introducing the cherry order graph $T$9 — whose $\binom{n}{2}$0 vertices are pseudo-cherries (all leaf pairs), with edges directed from pairs at shallower LCAs to deeper ones — the authors observe that $\binom{n}{2}$1 is acyclic. Any topological ordering of $\binom{n}{2}$2 translates into a sequence $\binom{n}{2}$3 of length $\binom{n}{2}$4, and an induction shows $\binom{n}{2}$5 reduces every non-trivial subtree of $\binom{n}{2}$6. Since every pseudo-cherry must appear somewhere in any such sequence, the bound is tight: the minimum length of a single sequence reducing all subtrees is exactly $\binom{n}{2}$7.

Non-finite covering numbers for networks

An appendix extends the framework to networks and reveals a sharp contrast with the tree case. For orchard networks, the orchard covering number $\binom{n}{2}$8 (minimum set of minimal cherry-picking sequences covering all subtrees) can be infinite: there exist orchard networks containing a subtree reducible by no sequence of the network, reproducing a known counterexample. Similarly, the tree-child covering number $\binom{n}{2}$9 can be infinite — the paper exhibits a tree-child network $c'(T)$0 and a subtree $c'(T)$1 such that no tree-child sequence of $c'(T)$2 reduces $c'(T)$3, because the forced first pick prevents $c'(T)$4 from ever appearing. This refines the known result that same-leaf-set subtrees are always reduced by any TCS: the guarantee fails once leaf sets differ. Interestingly, $c'(T)$5 is reducible by a non-tree-child cherry-picking sequence of $c'(T)$6, leaving open whether plain cherry-picking sequences suffice for all subtrees of tree-child networks.

Limitations and open questions

Several limitations are stated plainly. The recursive formula for $c'(T)$7 applies only to binary trees; for non-binary inputs only the survival variant is tractable, and the general covering number is characterized solely through equivalence to SubsetConnectivity for stars, with bounds verified only up to $c'(T)$8 computationally. Whether $c'(T)$9 occurs for some $\mathcal{T}_T$0 is unresolved. For networks, finiteness of the covering number is not guaranteed, and characterizing orchard networks with finite covering number is open. More broadly, the tractability of CoveringNumber for arbitrary combinations of network class and subnetwork family remains undetermined.

Conclusion

The paper settles CoveringNumber for binary phylogenetic trees through a clean product/sum recursion grounded in the survival covering number, extends that machinery to non-binary trees for the survival variant, and demonstrates that the full non-binary problem contains genuine combinatorial difficulty via its equivalence to SubsetConnectivity. The tight $\mathcal{T}_T$1 bound for single-sequence coverage and the non-finiteness results for tree-child and orchard networks delineate precisely where cherry-picking-based coverage is well-behaved and where it breaks down.