Tree Factorials on Rooted Metric Trees

• Tree factorials are sequences defined by a greedy minimization on rooted metric trees that generalize Bhargava’s logarithmic factorials.
• They employ a local edge-weighting process with recursive min–max formulas to capture branching structures and harmonic measures in random walks.
• Distinct interpretations in automata theory analyze decision-tree complexity for regular factorial languages, clarifying different notions of 'tree factorial'.

Tree factorial usually denotes the sequence of logarithmic factorials associated to a rooted tree, defined by a greedy minimization procedure on the boundary or, equivalently, by a local edge-weighting process. In this formulation, introduced for rooted metric trees, the sequence generalizes Bhargava’s factorials from subsets of local fields to arbitrary rooted trees and connects combinatorics on trees to random walks, unit flows, harmonic measure, and branching number (Amini, 2016). A distinct usage of closely related terminology appears in automata theory, where decision-tree complexity is studied for regular factorial languages; there, “tree” refers to decision trees rather than factorials of rooted trees (Moshkov, 2022).

1. Definition on rooted metric trees

Let $T$ be a rooted tree with root $t$, oriented away from $t$, and let $l:E(T)\to\mathbb{R}_+$ be a length function assigning each oriented edge $e$ its length $l(e)=l_e$. The metric realization of $(T,l)$ is the rooted metric tree $I_T$ obtained by gluing to each edge $e$ an open interval of length $l_e$ and identifying endpoints with vertices. The boundary $t$0 is the set of infinite oriented paths starting at $t$1, while the extended boundary $t$2 consists of $t$3 together with all finite oriented paths $t$4 from $t$5 to leaves $t$6 (Amini, 2016).

For $t$7, the intersection pairing is

$t$8

Fix any $t$9. Inductively, given $t$0, choose $t$1 arbitrarily among the unused elements of $t$2 that minimize

$t$3

and define

$t$4

The resulting sequence $t$5 depends only on the metric realization $t$6 of $t$7, not on the specific choices made during the greedy procedure. These numbers are called the $t$8-factorials, written $t$9; when $l:E(T)\to\mathbb{R}_+$0 is the standard length function, one writes $l:E(T)\to\mathbb{R}_+$1 (Amini, 2016).

For locally finite trees one may incorporate a capacity function $l:E(T)\to\mathbb{R}_+$2, constraining how many times a leaf may appear in the defining sequence. The factorials $l:E(T)\to\mathbb{R}_+$3 again depend only on the metric realization and the capacity function. A factorial-defining sequence is any sequence of boundary paths realizing the greedy construction (Amini, 2016).

2. Local weighting process and structural formulas

The greedy boundary definition admits an equivalent constructive formulation through a local weighting process on edges. A weighted tree is a pair $l:E(T)\to\mathbb{R}_+$4 with $l:E(T)\to\mathbb{R}_+$5 such that the edges of positive weight form a connected subtree $l:E(T)\to\mathbb{R}_+$6. A vertex is clear if all pending edges at that vertex are unweighted, and a vertex of $l:E(T)\to\mathbb{R}_+$7 is unsaturated if either it is an internal vertex of $l:E(T)\to\mathbb{R}_+$8 with at least one incident edge outside $l:E(T)\to\mathbb{R}_+$9, or it is a leaf of $e$0 whose incident edge has weight strictly below its capacity bound (Amini, 2016).

For a path $e$1, the weighted length is

$e$2

The process starts with $e$3, $e$4, and then iteratively selects an unsaturated vertex $e$5 minimizing $e$6, sets

$e$7

and updates weights along $e$8 and one or two strict descendant paths according to whether $e$9 is not clear, clear and branching, or a leaf with residual capacity. Theorem 2.1 states that the resulting sequence again depends only on $l(e)=l_e$0 and the metric tree, so the local weighting process is equivalent to the boundary greedy procedure (Amini, 2016).

The sequence terminates at the stopping index

$l(e)=l_e$1

where $l(e)=l_e$2 is the set of branching vertices and $l(e)=l_e$3 is the number of children of $l(e)=l_e$4. For a rooted locally finite tree with standard length, the integers $l(e)=l_e$5 for $l(e)=l_e$6 are the $l(e)=l_e$7-factorials (Amini, 2016).

A central structural formula is the recursive min–max relation. If the root has children $l(e)=l_e$8, with induced subtrees $l(e)=l_e$9, restricted length functions $(T,l)$0, restricted capacities $(T,l)$1, and stopping indices $(T,l)$2, then for all $(T,l)$3,

$(T,l)$4

This formula makes explicit how the factorials of a tree are assembled from the factorials of its rooted subtrees (Amini, 2016).

3. Relation to Bhargava’s factorials

Tree factorials were introduced as a combinatorial generalization of Bhargava’s logarithmic factorials. Let $(T,l)$5 be a local field with discrete valuation $(T,l)$6, valuation ring $(T,l)$7, maximal ideal $(T,l)$8, and residue field $(T,l)$9. For a subset $I_T$0, Bhargava’s logarithmic factorials are defined by choosing $I_T$1 and then choosing $I_T$2 minimizing

$I_T$3

with

$I_T$4

To such an $I_T$5 one associates its adelic tree $I_T$6: the vertices at height $I_T$7 are the images $I_T$8, with edges induced by the quotient maps $I_T$9 (Amini, 2016).

The basic identification is Proposition 1.2:

$e$0

Thus Bhargava’s factorials are exactly tree factorials of the adelic tree with standard lengths. This places ultrametric valuation geometry into the language of rooted trees and turns the valuation $e$1 into the tree intersection pairing $e$2 (Amini, 2016).

The capacity formalism extends this correspondence. For the ball $e$3 of radius $e$4 about the root in $e$5, define a capacity $e$6 on leaves by letting $e$7 be the cardinality of the extended boundary of the descendant subtree rooted at $e$8. Then

$e$9

the factorials of order $l_e$0 of $l_e$1. This shows that capacity-constrained tree factorials recover truncated arithmetic variants of Bhargava’s construction (Amini, 2016).

A plausible implication is that tree factorials serve as a common language for valuation-theoretic and purely combinatorial phenomena: the same greedy minimization principle can be interpreted either on a rooted tree boundary or on an infinite subset of a local field.

4. Asymptotic theory: random walk, harmonic measure, and branching number

The asymptotic behavior of normalized tree factorials is governed by potential theory on the tree. For an edge $l_e$2, define the conductance

$l_e$3

The random walk $l_e$4 started at the root moves from $l_e$5 to a neighbor $l_e$6 with probability

$l_e$7

The pair $l_e$8 is weakly complete if any infinite strict path consisting of degree-$l_e$9 vertices has infinite $t$00-length (Amini, 2016).

For an infinite locally finite rooted tree with weak completeness, Theorem 1.6 gives an exact criterion:

• $t$01 is transient;
• the limit

$t$02

is finite.

Equivalently, recurrence holds if and only if $t$03. The key analytic object is the unit current flow $t$04, the unique bounded-energy unit flow minimizing

$t$05

When the random walk is transient, the associated harmonic measure $t$06 on $t$07 satisfies

$t$08

where $t$09 is the cylinder set of rays passing through $t$10 (Amini, 2016).

Theorem 1.7 identifies the normalized factorial growth with the energy of the unit current flow:

$t$11

It also states that any factorial-defining sequence on $t$12 is equidistributed in $t$13 with respect to $t$14. Thus the greedy combinatorial procedure asymptotically samples the boundary according to harmonic measure, and its linear growth rate is the electrical energy of the network (Amini, 2016).

The same theory yields a factorial characterization of branching number. For $t$15, let

$t$16

Using Lyons’ criterion for transience of $t$17 and Theorem 1.6, Theorem 1.11 states

$t$18

This expresses branching number entirely in terms of finiteness of normalized factorial limits under an exponential deformation of the edge lengths (Amini, 2016).

5. Growth properties, examples, and computation

Tree factorials are superadditive:

$t$19

Consequently,

$t$20

exists in $t$21. In the transient weakly complete case one moreover has flow-based bounds

$t$22

for any bounded-energy unit flow $t$23, with equality for the unit current flow $t$24 (Amini, 2016).

Several examples illustrate the range of behaviors. For an infinite path, the extended boundary contains only a single infinite ray, so the greedy selection produces $t$25 and stops; hence $t$26 and only $t$27 is defined. If the path is finite and ends at a leaf $t$28 of total length $t$29, then for capacity $t$30 one has $t$31 and

$t$32

At the opposite extreme, for the regular $t$33-ary tree with standard lengths and $t$34, the simple random walk is transient, the current along an edge from level $t$35 to $t$36 is $t$37, and the energy is

$t$38

Therefore

$t$39

so asymptotically $t$40 (Amini, 2016).

Finite rooted trees yield terminating sequences. If the root is adjacent to $t$41 leaves and $t$42, then $t$43 and

$t$44

If the tree has two levels, with a root $t$45, one child $t$46, edge length $t$47, and two leaf children of $t$48 with $t$49, then $t$50 and

$t$51

These examples show that the factorial sequence can detect both branching and metric depth, but not uniquely determine the tree in general: different trees, including non-isomorphic metric trees, can share the same factorial sequence (Amini, 2016).

From an algorithmic perspective, the local weighting process supplies an explicit procedure for computing $t$52. Each iteration selects an unsaturated vertex minimizing weighted path length and updates weights along the root path and one or two strict descendant paths. The total number of iterations is $t$53, and a natural optimization is to maintain a priority structure keyed by the values $t$54 on unsaturated vertices (Amini, 2016).

6. Distinct automata-theoretic usage: decision trees for regular factorial languages

A separate line of work uses related terminology in a different sense. For a regular factorial language $t$55, one studies the depth of deterministic and nondeterministic decision trees whose queries reveal letters at positions of a word. Here a language is factorial if it is closed under taking factors, and the relevant objects are the layers

$t$56

Two tasks are considered: recognition, where the input is promised to belong to $t$57 and the objective is to identify the exact word, and membership, where the input is an arbitrary word of length $t$58 and the objective is to decide whether it belongs to $t$59 (Moshkov, 2022).

Rather than the minimum depth $t$60, the analysis uses the smoothed minimum depth

$t$61

which suppresses oscillations in $t$62 arising from parity or modular effects. For regular factorial languages, the asymptotic behavior is completely classified. In deterministic recognition, the smoothed depth is either $t$63, $t$64, or $t$65; in nondeterministic recognition and in deterministic and nondeterministic membership, it is either $t$66 or $t$67 (Moshkov, 2022).

The classification is phrased in terms of the $t$68-reduced source generating $t$69, especially the properties of simplicity, independence, and cyclic length. If the source is independent and simple with cyclic length at most $t$70, deterministic recognition has constant smoothed depth; if it is independent and simple with cyclic length at least $t$71, deterministic recognition has logarithmic smoothed depth; otherwise deterministic recognition is linear. Nondeterministic recognition is constant exactly in the independent simple case and linear otherwise. Membership is linear exactly when $t$72 is infinite and its complement is nonempty; otherwise it is constant (Moshkov, 2022).

This automata-theoretic notion is not a factorial sequence attached to a rooted tree. The shared vocabulary arises from factorial languages and decision trees, not from logarithmic factorials of tree boundaries. Distinguishing these two meanings avoids a common misconception: “tree factorial” in the sense of rooted-tree factorials concerns greedy intersection minimization, while the decision-tree literature concerns query complexity for regular factorial languages (Moshkov, 2022).