\section{Problem Setup} \label{sec:problem}

\textbf{Compositions come with tree.} The $d$ components of a composition are often organized by a known rooted tree $\mathcal{T}$, e.g., phylogenetic or taxonomic trees in ecology, gene ontologies in genomics \citep{ashburner2000gene, lozupone2005unifrac}. The tree encodes domain structure: which comparisons are meaningful and at what resolution. Hence, we seek a basis $V$ aligned with $\mathcal{T}$.

\textbf{Binary trees.}
When $\mathcal{T}$ is binary, each internal node $u$ has exactly two children, yielding one \emph{contrast}: the log-ratio of geometric means of the two descendant clades. This is a special case of sequential binary partitioning (SBP) \citep{egozcue2005groups}, which \citet{silverman2017phylogenetic} applied to phylogenies as \emph{PhILR}. PhILR is well-matched to that setting, as phylogenies are typically inferred as bifurcating, and here, orthonormality is straightforward: contrasts at disjoint nodes have disjoint support, and contrasts at nested nodes are orthogonal because the inner contrast sums to zero on each child clade. This construction works because binary branching imposes minimal local structure: each node requires exactly one contrast, yielding a one-to-one correspondence between internal nodes and basis vectors. The global consistency problem asking that local contrasts compose into an orthonormal basis on leaves reduces to verifying {\em pairwise} orthogonality, which holds by support structure and zero-sum constraints.

\textbf{What happens with polytomies?}
For general trees $\mathcal{T}$, the simplicity above breaks down (Figure~\ref{fig:binary-vs-poly}). Consider a node $u$ with $k_u > 2$ children. Comparing $k_u$ clades requires $k_u - 1$ orthogonal contrasts---a subspace, not a single vector. Several challenges arise: (i) defining canonical local contrasts at $u$, (ii) extending them to global vectors on leaves, and (iii) ensuring orthogonality across all nodes. Standard approaches fail because subtrees of different sizes contribute unequally to inner products, as detailed shortly.

{\color{black}
\textbf{Polytomies are common in practice.}
Polytomies arise within phylogenies as both hard polytomies (e.g., rapid radiations) and soft polytomies (e.g., collapsed low-support nodes) \citep{maddison1989reconstructing}. About 64\% of taxonomic branch points in the NCBI Taxonomy Database have three or more children \citep{lin2011polytomy} and biomedical ontologies routinely encode multi-way groupings, e.g., the cell ontology \citep{diehl2016cell}. Such curated trees often carry \emph{meaningful internal structure} (e.g., independently named internal nodes) that can inform downstream representations. 
}
%
{However, in practice, one typically \emph{arbitrarily refines} them into binary trees \citep{lin2011polytomy}. This introduces additional internal nodes and splits {\em not present in the original hierarchy}, making resulting coordinates and interpretations binarization-dependent. Alternative approaches aim to identify predictive log-ratio features via log-contrast regression, greedy balance selection \citep{rivera2018balances}, pairwise log-ratio testing \citep{mandal2015analysis}, or phylogeny-guided ILR factors via edge selection \citep{washburne2017phylogenetic}. But these methods yield isolated contrasts rather than a full, node-grouped orthonormal coordinate system canonically tied to a given multifurcating tree.
}

% Polytomies are common in practice; 64\% of branch points in the NCBI Taxonomy Database have three or more children \citep{lin2011polytomy}. Practitioners typically \emph{arbitrarily refine} them into binary trees \citep{lin2011polytomy}. This introduces additional internal nodes and splits {\em not present in the original hierarchy}, making resulting coordinates and interpretations binarization-dependent. Alternative approaches aim to identify predictive log-ratio features via log-contrast regression, greedy balance selection \citep{rivera2018balances}, pairwise log-ratio testing \citep{mandal2015analysis}, or phylogeny-guided ILR factors via edge selection \citep{washburne2017phylogenetic}. But these methods yield isolated contrasts rather than a full, node-grouped orthonormal coordinate system canonically tied to a given multifurcating tree.

No existing method provides a canonical, complete orthonormal decomposition for general trees that simultaneously defines local contrasts, extends them globally, and ensures consistency. Obtaining such a decomposition without sacrificing isometry or introducing arbitrary choices is our goal.
