[Paper Review] The mean value of the squared path-difference distance for rooted phylogenetic trees

Arnau Mir, Francesc Rosselló|ArXiv.org|Jun 13, 2009

Evolution and Paleontology Studies12 references3 citations

TL;DR

This paper computes the mean squared path-difference distance between two fully resolved rooted phylogenetic trees with $n$ leaves under the uniform distribution, deriving an exact formula that grows as $O(n^3)$. The key result shows this mean value equals the corresponding value for unrooted trees with $n+1$ leaves, revealing a deep combinatorial relationship between rooted and unrooted tree metrics.

ABSTRACT

The path-difference metric is one of the oldest distances for the comparison of fully resolved phylogenetic trees, but its statistical properties are still quite unknown. In this paper we compute the mean value of the square of the path-difference metric between two fully resolved rooted phylogenetic trees with $n$ leaves, under the uniform distribution. This complements previous work by Steel and Penny, who computed this mean value for fully resolved unrooted phylogenetic trees.

Motivation & Objective

• To determine the statistical properties of the path-difference metric for rooted phylogenetic trees, which remain largely unknown despite its long-standing use.
• To compute the expected value of the squared path-difference distance between two uniformly random fully resolved rooted trees with $n$ leaves.
• To establish a quantitative relationship between rooted and unrooted tree metrics, particularly in terms of mean squared path-difference.
• To provide a benchmark for assessing tree similarity, enabling researchers to evaluate whether observed tree differences are statistically significant.

Proposed method

• The authors use combinatorial enumeration techniques to compute the sum of squared path differences across all pairs of rooted trees.
• They derive closed-form expressions for $S_n^{(1)} = \sum_{T \in \mathcal{T}_n} d_T(1,2)$ and $S_n^{(2)} = \sum_{T \in \mathcal{T}_n} d_T(1,2)^2$, representing first and second moments of path lengths.
• The method relies on hypergeometric series and known combinatorial identities, particularly those involving double factorials and central binomial coefficients.
• The derivation leverages symmetry and uniform distribution over the set $\mathcal{T}_n$ of fully resolved rooted trees with $n$ leaves.
• The final formula for the mean squared path-difference distance is derived by combining the first and second moments and simplifying using known identities.
• The authors use Stirling's approximation to analyze asymptotic behavior, showing the mean value grows as $O(n^3)$.

Experimental results

Research questions

• RQ1 What is the expected value of the squared path-difference distance between two uniformly random fully resolved rooted phylogenetic trees with $n$ leaves?
• RQ2 How does the mean squared path-difference distance for rooted trees compare to that for unrooted trees of the same or different size?
• RQ3 Can a closed-form expression be derived for the mean squared path-difference metric in the rooted case, given the complexity introduced by the root's asymmetric role?
• RQ4 What is the asymptotic growth rate of the mean squared path-difference distance as $n$ increases?

Key findings

• The mean squared path-difference distance for rooted trees with $n$ leaves is $2\binom{n}{2}\left(4(n-1)+2 - \frac{2^{2(n-1)}}{\binom{2(n-1)}{n-1}} - \left(\frac{2^{2(n-1)}}{\binom{2(n-1)}{n-1}}\right)^2\right)$.
• This value grows asymptotically as $O(n^3)$, consistent with the asymptotic behavior of the unrooted case.
• The mean squared path-difference for rooted trees with $n$ leaves equals the corresponding value for unrooted trees with $n+1$ leaves.
• The mean path length between two random leaves in a rooted tree is $\frac{2^{2(n-1)}}{\binom{2(n-1)}{n-1}}$, which matches the unrooted case with $n-1$ leaves.
• The derived formula provides a statistical baseline for assessing tree similarity, enabling researchers to test whether observed differences are significant.
• The result confirms a deep combinatorial link between rooted and unrooted tree metrics, despite the lack of a direct structural explanation.