[Paper Review] On the neighbourhoods of trees

Peter Humphries, Taoyang Wu|arXiv (Cornell University)|Feb 10, 2012

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TL;DR

This paper provides a closed-form expression for the size of the tbr (tree bisection and reconnection) neighbourhood of a phylogenetic tree, resolving a long-standing question about how many distinct trees can be reached via a single tbr operation. The authors derive the formula based on tree topology, particularly the distribution of subtrees and clusters, and establish that the neighbourhood size depends critically on tree shape, with caterpillar trees maximizing the size and complete binary trees minimizing it asymptotically.

ABSTRACT

Tree rearrangement operations typically induce a metric on the space of phylogenetic trees. One important property of these metrics is the size of the neighbourhood, that is, the number of trees exactly one operation from a given tree. We present an expression for the size of the TBR (tree bisection and reconnection) neighbourhood, thus answering a question first posed in [Annals of Combinatorics, 5, 2001 1-15].

Motivation & Objective

• To resolve the open problem of determining the exact size of the tbr neighbourhood for any unrooted binary phylogenetic tree.
• To characterize how tree shape—specifically the distribution of clusters and subtree sizes—affects the number of trees reachable via a single tbr operation.
• To establish tight asymptotic upper and lower bounds for the tbr neighbourhood size across all binary tree topologies in the space $\mathscr{T}_n$.
• To reprove known results for the nni and spr neighbourhoods using a unified framework based on cluster decomposition and partial splits.
• To provide a general expression for the tbr neighbourhood size that depends on the binary expansion of the number of leaves and the tree's hierarchical structure.

Proposed method

• Define tbr operations as edge deletions followed by reconnection with subdivision to preserve binary structure, ensuring the resulting tree is distinct and valid.
• Use the concept of partial splits and clusters to decompose the tree into subtrees of specific sizes, particularly focusing on subtrees of size $2^j$.
• Introduce a function $\tau(m)$ to detect whether a subtree of size $m$ contains a cluster of size $2^j$, which affects the count of valid tbr operations.
• Apply a recursive decomposition based on the binary representation of $n$, the number of leaves, to compute the total number of tbr operations via summation over subtree sizes.
• Derive a general formula for $\Gamma(\mathcal{T})$, the total number of tbr operations, using the coefficients $\alpha_i$ in the binary expansion of $n$ and the structure of the tree’s cluster decomposition.
• Use Theorem 3.6 to relate $\Gamma(\mathcal{T})$ to the final tbr neighbourhood size $|N_{\textsc{tbr}}(\mathcal{T})|$ via a transformation involving $n^2$ and linear terms.

Experimental results

Research questions

• RQ1 What is the exact size of the tbr neighbourhood for a given unrooted binary phylogenetic tree, and how does it vary with tree topology?
• RQ2 How do the structural properties of a tree—such as the presence of cherries, clusters, and subtree sizes—affect the number of tbr operations possible?
• RQ3 What tree shapes maximize or minimize the size of the tbr neighbourhood across $\mathscr{T}_n$?
• RQ4 Can the tbr neighbourhood size be expressed as a closed-form formula based on the binary representation of $n$ and the tree’s hierarchical cluster structure?
• RQ5 What are the tight asymptotic upper and lower bounds for the tbr neighbourhood size over all binary trees with $n$ leaves?

Key findings

• The paper provides a closed-form expression for $|N_{\textsc{tbr}}(\mathcal{T})|$ that depends on the tree’s topology, particularly the binary decomposition of $n$ and the presence of subtrees of size $2^j$.
• For a perfect tree with $n = 3 \cdot 2^{k-1}$, the tbr neighbourhood size is $|N_{\textsc{tbr}}(\mathcal{T})| = n^2(4k - \frac{32}{3}) + 22n - 6$.
• For a perfect tree with $n = 2^k$, the size is $|N_{\textsc{tbr}}(\mathcal{T})| = n^2(4k - 13) + 22n - 6$.
• For complete binary trees, the tbr neighbourhood size satisfies $|N_{\textsc{tbr}}(\mathcal{T})| = 4n^2\lfloor \log_2 n \rfloor + O(n^2)$, establishing an asymptotically tight lower bound.
• The maximum tbr neighbourhood size is achieved by caterpillar trees, matching the previously known upper bound of $\frac{2}{3}n^3 - 4n^2 + \frac{16}{3}n + 2$.
• The minimum tbr neighbourhood size occurs for complete binary trees, and the asymptotic growth is $\Theta(n^2 \log n)$, confirming the tightness of the lower bound.