Forest-Representable Matrices

• Forest-representable matrices are defined by block structures that follow forest (or tree) topologies, enabling sparse or structured inverses and efficient low-rank factorizations.
• They support fast algebraic operations like matrix-vector products and inversions by leveraging tree quasi-separable constructions and nested bases.
• Applications span numerical linear algebra and combinatorial matrix theory, generalizing formats like SSS and HSS while also connecting to Laplacian and total positivity formulations.

Forest-representable matrices are matrices whose structure is governed by a forest. In the rank-structured formulation introduced for tree quasi-separable matrices, a matrix $A \in \mathbb{R}^{n\times n}$ is forest-representable if there exists a block partition of its rows and columns indexed by the vertices of a forest $F=\bigsqcup_c T_c$ such that either $A$ has block-sparsity consistent with $F$, or $A^{-1}$ has block-sparsity consistent with $F$; when the forest has multiple components, a permutation puts $A$ into block-diagonal form, one block per tree component (Govindarajan et al., 2024). In other settings, closely related terminology is used for matrices whose entries enumerate forests, matrices derived from spanning rooted forests, adjacency-based encodings that classify forests, and lower-triangular matrices that encode ranked tree shapes (Chebotarev et al., 2013, Traldi, 2020, Jennings-Shaffer et al., 30 Oct 2025). This suggests that the term does not denote a single universal matrix class, but rather a family of matrix constructions in which forest topology is the organizing constraint.

1. Forest topology as a matrix constraint

Let $F=(V,E)$ be an undirected acyclic graph, possibly disconnected, with connected components $T_c=(V_c,E_c)$, and let each vertex $v\in V$ carry an index set $F=\bigsqcup_c T_c$0. In the numerical linear-algebraic sense, forest-representability means that $F=\bigsqcup_c T_c$1 is viewed as a graph-partitioned block matrix $F=\bigsqcup_c T_c$2, and either $F=\bigsqcup_c T_c$3 unless $F=\bigsqcup_c T_c$4 or $F=\bigsqcup_c T_c$5, or the same sparsity statement holds for $F=\bigsqcup_c T_c$6 instead of $F=\bigsqcup_c T_c$7 (Govindarajan et al., 2024).

The disconnected case is not merely formal. If $F=\bigsqcup_c T_c$8 has several components, the permutation bringing $F=\bigsqcup_c T_c$9 to forest form yields a block-diagonal matrix with blocks indexed by the connected components $A$0. Each block can then be represented and operated on independently, so a forest-representable matrix is a direct sum of tree quasi-separable blocks, one per tree component (Govindarajan et al., 2024).

A closely related situation arises for sparse matrices whose adjacency graph is itself a tree. Such a sparse matrix $A$1 is sparse, whereas $A$2 is typically dense; nevertheless, $A$3 satisfies the graph-induced rank structure and admits an exact tree quasi-separable form with generator sizes equal to the ranks of unit Hankel blocks (Govindarajan et al., 2024). In this sense, forest-representability includes both explicit forest sparsity and forest-shaped inverse sparsity.

2. Tree quasi-separable realization

For a rooted tree $A$4 with root $A$5 and a partition $A$6 of $A$7, a tree quasi-separable (TQS) matrix is defined by local generators attached to vertices and edges. These include the explicit diagonal blocks $A$8, input-to-edge maps $A$9 and $F$0, edge-to-edge transfers $F$1, $F$2, $F$3, and edge-to-output maps $F$4 and $F$5 (Govindarajan et al., 2024).

For $F$6, the off-diagonal block is obtained by multiplying operators along the unique path $F$7: $F$8 This path factorization implies that off-diagonal blocks between disjoint subtrees are low-rank and can be written in the canonical form

$F$9

where $A^{-1}$0 and $A^{-1}$1 are bases assembled from local generators and $A^{-1}$2 collects the couplings transmitted along the path and sibling transfers (Govindarajan et al., 2024).

An important structural feature is nestedness. Internal-node bases are formed from child bases through small transfer matrices: $A^{-1}$3 Diagonal and near-diagonal blocks remain explicit through the $A^{-1}$4, while compressed representation is reserved for off-diagonal interactions connecting disjoint subtrees across the tree topology (Govindarajan et al., 2024).

3. Hankel-rank characterization and specialization to SSS and HSS

The decisive structural invariant in the TQS theory is the rank of tree-induced Hankel blocks. For an edge $A^{-1}$5, one defines a unit Hankel subset by

$A^{-1}$6

where $A^{-1}$7 denotes the descendants of $A^{-1}$8, including $A^{-1}$9 itself, and the associated Hankel block is $F$0 (Govindarajan et al., 2024).

The key theorem states that if $F$1 is a graph-partitioned matrix on a tree, then $F$2 admits a TQS representation whose rank profile satisfies

$F$3

and this realization is minimal: any other TQS realization on the same tree has edge ranks $F$4 for all $F$5 (Govindarajan et al., 2024). The intuition is that each unit Hankel block measures the minimal state dimension needed to transmit interactions across the corresponding cut; any smaller edge state would force a lower-rank factorization of $F$6, which is impossible.

This theorem simultaneously recovers the two classical semi-separable formats. When the tree is a path graph, rooted at the last node, the TQS operators reduce to the classical sequentially semi-separable representation; the relevant Hankel blocks coincide with the classical SSS off-diagonal Hankel blocks, and their ranks determine the generator sizes (Govindarajan et al., 2024). When the tree is a post-ordered binary partition tree with internal nodes taken to be empty, the TQS form becomes hierarchically semi-separable, with sibling interactions compressed level by level and nested bases propagated across levels (Govindarajan et al., 2024).

A common misconception is that forest-representability in this setting is merely a notational reformulation of SSS or HSS. The TQS construction is instead a simultaneous generalization: the path graph yields SSS, the hierarchical binary tree yields HSS, and general trees interpolate between these two extremes (Govindarajan et al., 2024).

4. Algebraic operations, conversion algorithms, and efficiency regime

The TQS class is closed under addition, multiplication, and inversion, provided the input and output partitions coincide in the inversion case. If $F$7 is the maximum generator rank and $F$8 is the maximum node degree, the associated algorithms are fast when $F$9 is small, $A$0 is bounded, and the block sizes $A$1 are uniformly bounded (Govindarajan et al., 2024).

Operation	Structural statement	Stated cost
Matrix–vector product	state-space upsweep and downsweep	$A$2 flops
Addition	$A$3	edge-wise concatenation and recompression $A$4
Multiplication	$A$5	$A$6 per mat-vec and $A$7 for recompression
Inversion / direct solve	$A$8 for nonsingular $A$9	typical direct solve $F=(V,E)$0

Conversion from a dense matrix to TQS proceeds in two passes. In the upsweep, for each edge $F=(V,E)$1 with $F=(V,E)$2, one forms the unit Hankel block $F=(V,E)$3, computes a low-rank factorization $F=(V,E)$4, assigns $F=(V,E)$5 and $F=(V,E)$6, and compresses interior augmented matrices to obtain transfer operators. In the downsweep, for each edge $F=(V,E)$7 with $F=(V,E)$8, one analogously factors $F=(V,E)$9, assigns $T_c=(V_c,E_c)$0 and $T_c=(V_c,E_c)$1, and extracts the sibling and parent transfers $T_c=(V_c,E_c)$2 and $T_c=(V_c,E_c)$3 (Govindarajan et al., 2024).

The factorization at each edge may use deterministic truncated SVD, strong RRQR, or randomized SVD. Accuracy is controlled by a truncation tolerance $T_c=(V_c,E_c)$4; if each unit Hankel block $T_c=(V_c,E_c)$5 is approximated with error $T_c=(V_c,E_c)$6, then the reconstruction error obeys

$T_c=(V_c,E_c)$7

with $T_c=(V_c,E_c)$8 depending mildly on $T_c=(V_c,E_c)$9 and path lengths (Govindarajan et al., 2024).

The format is not universally economical. For arbitrary dense matrices, the unit Hankel blocks can have large rank, yielding large generators. The arrowhead matrix on a star tree is the standard counterexample: the TQS representation then uses roughly as many parameters as the dense matrix, while an SSS or HSS representation on a different topology may be more efficient (Govindarajan et al., 2024).

5. Graph-theoretic, Laplacian, and matroidal formulations

Several other matrix constructions encode forests directly rather than through low-rank path factorization. A real symmetric matrix is called acyclic if it is the adjacency matrix of a weighted forest with real nonzero edge weights and possibly vertex weights; if the matrix is irreducible, the forest is a tree (Bahmani et al., 2016). For such matrices, the characteristic polynomial equals the matching polynomial of the weighted forest, and explicit eigenvector formulas are available in terms of path weights and characteristic polynomials of vertex- or path-deleted submatrices (Bahmani et al., 2016).

A different line of work studies Laplacian-based matrices of spanning rooted forests. For a weighted digraph with column Laplacian $v\in V$0, the matrix-forest theorem gives

$v\in V$1

where the entries of $v\in V$2 and $v\in V$3 admit combinatorial interpretations in terms of weighted spanning out-forests (Chebotarev et al., 2013). The limiting matrix $v\in V$4 is the eigenprojection of $v\in V$5 onto its nullspace, satisfies $v\in V$6, and exposes the source-knot structure of the digraph (Chebotarev et al., 2013).

Forests can also be recovered from adjacency-derived binary matroids. Over $v\in V$7, the matrices

$v\in V$8

define the restricted isotropic and isotropic matroids of a forest $v\in V$9. For forests $F=\bigsqcup_c T_c$00 and $F=\bigsqcup_c T_c$01, the equivalence

$F=\bigsqcup_c T_c$02

shows that these matrix representations classify forests up to isomorphism (Traldi, 2020).

Hierarchical trees admit yet another sparse matrix representation, the Generation Matrix. Under descending order of height, the Generation Matrix is lower triangular, has exactly $F=\bigsqcup_c T_c$03 nonzeros for a tree with $F=\bigsqcup_c T_c$04 nodes, and becomes block-diagonal for forests, one block per component (Cai et al., 2022). This representation is designed so that triangular solves simulate upward and downward recursive traversals (Cai et al., 2022).

6. Enumerative, positivity, and ranked-tree encodings

In combinatorics, lower-triangular matrices whose entries enumerate trees and forests are also described through forest structure. The matrix

$F=\bigsqcup_c T_c$05

counts rooted labeled trees on $F=\bigsqcup_c T_c$06 with a prescribed root statistic, and also partial functional digraphs on $F=\bigsqcup_c T_c$07 with exactly $F=\bigsqcup_c T_c$08 vertices of out-degree $F=\bigsqcup_c T_c$09; its row-generating polynomials are $F=\bigsqcup_c T_c$10, the matrix is totally positive, and the sequence $F=\bigsqcup_c T_c$11 is coefficientwise Hankel-totally positive (Chen et al., 2023). The related forest matrix

$F=\bigsqcup_c T_c$12

enumerates $F=\bigsqcup_c T_c$13-component forests of rooted trees on $F=\bigsqcup_c T_c$14, with weighted generalizations by improper edges and proper-child multiplicities; its matrix and row-generating polynomials are coefficientwise totally positive and coefficientwise Hankel-totally positive under Toeplitz-total positivity assumptions on the weight sequence $F=\bigsqcup_c T_c$15 (Sokal, 2021). A $F=\bigsqcup_c T_c$16-analogue replaces binomial coefficients and powers by Gaussian binomials and $F=\bigsqcup_c T_c$17-integers, yielding coefficientwise totally positive matrices proved via planar networks and the Lindström–Gessel–Viennot lemma (Gilmore, 2021).

Restricted Stirling and Lah number matrices provide another forest-based representation. For restriction sets $F=\bigsqcup_c T_c$18 with $F=\bigsqcup_c T_c$19, each entry of the inverse matrices $F=\bigsqcup_c T_c$20, $F=\bigsqcup_c T_c$21, and $F=\bigsqcup_c T_c$22 is a signed difference of cardinalities of explicitly defined families of increasingly ordered, min-first ordered, or linearly ordered phylogenetic forests; when $F=\bigsqcup_c T_c$23 has no exposed odds, each inverse entry is, up to an explicit sign, the cardinality of a single family of $F=\bigsqcup_c T_c$24-good forests (Engbers et al., 2016).

A more recent phylogenetic use of forest-representable matrices is the $F=\bigsqcup_c T_c$25-matrix for ranked tree shapes. For an $F=\bigsqcup_c T_c$26-leaf fully heterochronous ranked tree shape, the $F=\bigsqcup_c T_c$27-matrix is lower triangular of size $F=\bigsqcup_c T_c$28, and $F=\bigsqcup_c T_c$29 counts the number of edges whose parent rank is at most $F=\bigsqcup_c T_c$30 and whose child rank is greater than $F=\bigsqcup_c T_c$31 (Jennings-Shaffer et al., 30 Oct 2025). Valid $F=\bigsqcup_c T_c$32-matrices are characterized by monotonicity constraints, diagonal and subdiagonal rules, and a four-entry local bound

$F=\bigsqcup_c T_c$33

with $F=\bigsqcup_c T_c$34 and $F=\bigsqcup_c T_c$35 determined by the left, above, above-left, and previous diagonal values (Jennings-Shaffer et al., 30 Oct 2025). This local rule yields an explicit bijection with ranked tree shapes and supports enumeration and probabilistic modeling (Jennings-Shaffer et al., 30 Oct 2025).

Across these settings, forest-representable matrices serve different purposes: low-rank compression for sparse-inverse problems, Laplacian resolvents for spanning forests, algebraic classification of forest graphs, total positivity in forest enumerators, and lower-triangular encodings of ranked tree shapes. The shared principle is precise rather than metaphorical: a forest supplies the combinatorial skeleton that determines which matrix entries are allowed, how they factor, or what they count.